Periodic Locally Compact Groups: A Study of a Class of Totally Disconnected Topological Groups 9783110599190, 9783110598476

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Wolfgang Herfort, Karl H. Hofmann, and Francesco G. Russo Periodic Locally Compact Groups

De Gruyter Studies in Mathematics

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Edited by Carsten Carstensen, Berlin, Germany Gavril Farkas, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesztesy, Waco, Texas, USA Niels Jacob, Swansea, United Kingdom Zenghu Li, Beijing, China Karl-Hermann Neeb, Erlangen, Germany

Volume 71

Wolfgang Herfort, Karl H. Hofmann, and Francesco G. Russo

Periodic Locally Compact Groups |

A Study of a Class of Totally Disconnected Topological Groups

Mathematics Subject Classification 2010 Primary: 22A05, 22A26; Secondary: 20E34, 20K35 Authors Prof. Dr. Wolfgang Herfort Institut for Analysis & Scientific Computing Vienna University of Technology Wiedner Hauptstr. 8-10 1040 Vienna Austria [email protected] Prof. Dr. Karl H. Hofmann Fachbereich Mathematik Technische Universität Darmstadt Schloßgartenstr. 7 64289 Darmstadt Germany [email protected] and Department of Mathematics Tulane University LA 70118 New Orleans USA [email protected]

Prof. Dr. Francesco G. Russo Department of Mathematics and Applied Mathematics University of Cape Town Private Bag X1, Rondebosch Cape Town 7701 South Africa [email protected]

ISBN 978-3-11-059847-6 e-ISBN (PDF) 978-3-11-059919-0 e-ISBN (EPUB) 978-3-11-059908-4 ISSN 0179-0986 Library of Congress Control Number: 2018951343 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2019 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

This book is dedicated to

Anna Herfort, Isolde Hofmann, and Licia Riccitelli

Preface What does it mean for a group if all of its subgroups are normal? And what, short of that, if at least for all subgroups A and B the set-product AB is again a subgroup, that is, AB = BA? These questions were asked and answered at an early stage in the development of group theory. A comprehensive source of information is the book by Roland Schmidt.1 Naturally, corresponding questions were asked for topological and, in particular, for locally compact groups G and for subgroups taken form the set 𝒮𝒰ℬ(G) of all closed subgroups. The literature exhibits quite a number of forays into this field, at least in the domain of compact topological groups. Given the state of knowledge on locally compact groups one would rightfully assume that they present a reasonable territory for a systematic approach to questions of this kind. However, a comprehensive sourcebook for such an approach has not yet been proposed. The present book deals with periodic locally compact groups. A locally compact group G is called periodic, if it satisfies the following two properties: – there are no nonsingleton connected subspaces in G, – every element in G is contained in a compact subgroup. All profinite groups satisfy these conditions, and beyond compact groups they are satisfied by the p-adic completions ℚp of the field ℚ of rational numbers, and accordingly, by finite-dimensional ℚp -vector spaces and their automorphism groups. The reason for dealing with this category of locally compact groups is that it provides us with a wealth of tools that we use in order to work at length with a class of locally compact groups which we call near abelian for reasons that will become clear as soon as we present our definition. And why, in the end, do we concentrate exactly on this class of solvable locally compact groups? The answer is this: Indeed within the domain of near abelian groups all the group theoretical properties that we hinted at in the beginning of this preface can be uniformly and systematically treated and explained appropriately in a topological-algebraic fashion. Through this strategy, our approach differs from what is found in the literature so far. In our approach we – develop the Sylow and Hall theory of locally compact periodic groups, – view contemporary aspects of the compact Hausdorff space defined on 𝒮𝒰ℬ(G), nowadays called the Chabauty space, of G, 1 Roland Schmidt, “Subgroup Lattices of Groups,” de Gruyter Expositions in Mathematics 14, Berlin, 1994. https://doi.org/10.1515/9783110599190-201

VIII | Preface –

–

dwell on the number theoretical and graph theoretical subtleties of “scalar multiplication” of periodic locally compact abelian groups and clarify them as completely as possible, and indeed re-evaluate some aspects of locally compact abelian groups themselves that are neither commonly known nor easily accessible.

Obviously, we have to traverse fields of technical complications. Therefore we decided that it would be helpful for readers and users of this text to have the main layout and its essential results available in an overview chapter that is not interrupted and slowed down by the details of the proofs and their systematics. And so the first thing that the reader will encounter is such an overview of the principal content of the book. The beginning of the joint work on this study may be traced back to papers by Francesco G. Russo and Karl H. Hofmann2 on the likelihood that two elements commute in a compact group. These findings can in turn be found in the third edition of the monograph “The Structure of Compact Groups,” pp. 537–544.3 They led to the introduction of “near abelian” groups in the framework of profinite p-groups.4 It was Wolfgang Herfort who, after evaluating this material, insisted that this concept required a more general background. His persistence moved the accent of the study away from its concentration on profinite groups to the much wider field of arbitrary locally compact groups and so opened it to many noteworthy links with classical group theory. As far as the history of our topic is concerned we have tried hard to cover it amply in the course of our text. In writing on our subject, we had to inspect a considerable body of literature, not all of which is eventually referenced in the text itself. Nevertheless, we mention some such pertinent sources in our bibliography. The reader may take their listing as an encouragement for additional reading. We acknowledge with gratitude the support of many persons and institutions. Wolfgang Herfort is grateful to his home institution, the Institute of Analysis and Scientific Computation at the Technische Universität Wien, and to the Mathematics Department of the Brigham Young University during the year 2015; a significant part of the present work has been accomplished during this time. Karl Heinrich Hofmann is thankful for the scholarly support he received through the years both from the Technische Universität Darmstadt and from Tulane University in New Orleans. Portions of this text were completed while Karl Heinrich Hofmann and Linus Kramer stayed at the Mathematisches Forschungsinstitut Oberwolfach in the Program Research in Pairs 2 Karl H. Hofmann and Francesco G. Russo, The probability that x and y commute in a compact group, Math. Proc. of the Cambridge Phil. Soc. 153 (2012), 557–571. 3 Karl H. Hofmann and Sidney A. Morris, “The Structure of Compact Groups,” 3rd Edition, De Gruyter Studies in Mathematics 25, Berlin, 2013, pp. xxii+924. 4 Karl H. Hofmann and Francesco G. Russo, Near abelian profinite groups, Forum Mathematicum 27 (2015), 647–698.

Preface

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in February 2017 and February 2018, which also permitted intensive electronic correspondence with Wolfgang Herfort in Vienna. Karl Heinrich Hofmann also gratefully acknowledges numerous illuminating conversations with Rafael Dahmen at Technische Universität Darmstadt on many subjects in this book. The wise guidance of the experienced writer Sidney A. Morris is noted with gratitude. Francesco G. Russo gratefully acknowledges support from NRF of South Africa with Ref. No. 93652. The authors are indebted to the organizers of the Workshop on Topology and Topological Groups in December 2017 at the African Institute for Mathematical Sciences. The contents of this monograph were introduced at the conference. Finally we are thankful to the De Gruyter for making accessible the helpful SkyLaTex system and the overall patience of Ina Talandienė and Ieva Spudulytė, and, the LATEX-nical support. July 2018 Wolfgang Herfort Technische Universität Wien Karl Heinrich Hofmann Technische Universität Darmstadt and Tulane University, New Orleans Francesco G. Russo University of Cape Town

Contents Preface | VII Overview | XV

Part I: Background information on locally compact groups Introduction | 3 1 1.1 1.2 1.2.1 1.3

Locally compact spaces and groups | 5 The presence of many compact open subgroups | 5 The hyperspace of closed subgroups 𝒮𝒰ℬ(G) of G | 9 The Chabauty space of a compactly ruled group | 11 Semidirect products | 13

2 2.1 2.2 2.3 2.4 2.5 2.6

Periodic locally compact groups and their Sylow theory | 19 Normal σ-subgroups | 22 Normal σ-Sylow subgroups | 25 A Schur–Zassenhaus theorem | 26 The fixed point theorem | 33 Pairwise commuting p-Sylow subgroups | 38 Sylow bases in inductively prosolvable groups | 43

3 3.1 3.2 3.3 3.3.1

Abelian periodic groups | 47 Braconnier’s theorem | 47 Preliminaries about the p-rank | 50 Locally compact abelian torsion groups | 52 A nonsplit extension of a reduced locally compact abelian p-group by ℚp | 55 Purity used partially | 67 Locally compact abelian divisible groups | 69 Torsion-freeness and divisibility in p-groups | 72 Splitting in torsion-free p-groups | 78 The largest divisible subgroup | 79 Dense divisible subgroups | 83 Nonsplitting of ℚp in the presence of torsion | 86 Divisible torsion groups | 91 The p-rank of a locally compact abelian p-group | 93 Structure of locally compact p-groups of finite p-rank | 96

3.4 3.5 3.6 3.6.1 3.6.2 3.7 3.8 3.9 3.10 3.11

XII | Contents 4 4.1 4.1.1 4.1.2 4.2 4.3 4.3.1 4.3.2 4.3.3 4.3.4 4.3.5 5 5.1 5.1.1 5.1.2 5.2 5.3 5.4

Scalar automorphisms and the mastergraph | 101 On scalar automorphisms | 103 The structure of ℤp for p ≠ 2 | 105 The structure of ℤ2 | 108 The structure of the group of scalar automorphisms | 110 A bipartite graph for scalar action on a periodic locally compact abelian group | 114 Geometric properties of the mastergraph | 115 ̃ × and the prime mastergraph | 116 The structure of ℤ The Sylow decomposition of ℤ(n)× indexed by 𝒢 | 118 The structure of SAut(A) and its prime graph 𝒢 (A) | 119 The structure of a scalar action and its prime graph | 121 Inductively monothetic groups | 127 Classifying inductively monothetic subgroups | 128 The build-up of Π-procyclic groups from open procyclic subgroups | 130 Inductively monothetic groups and divisibility | 131 The subspace of inductively monothetic subgroups | 137 On divisibility and ℤp | 138 The extensions of a locally compact group by a Π-procyclic group | 139

Part II: Near abelian groups Introduction | 149 6 6.1

The definition of near abelian groups | 151 Basic definitions | 151

7 7.1 7.2 7.3 7.4 7.5 7.6

Important consequences of the definitions | 157 Periodicity of near abelian groups | 157 Near abelian p-groups | 159 Nontrivial near abelian groups | 160 A first glance at the Sylow theory of periodic near abelian groups | 163 The existence of scaling subgroups | 167 Singular near abelian groups | 176

8 8.1 8.2

Trivial near abelian groups | 179 The subset B(G) in a trivial near abelian group | 182 General structure of trivial near abelian groups | 186

Contents | XIII

9 9.1 9.2 9.3 9.4

The class of near abelian groups | 191 Closed subgroups | 191 Quotient groups | 193 Projective limits | 194 Inductive limits | 196

10

The Sylow structure of periodic nontrivial near abelian groups and their prime graphs | 203 More on the Sylow structure of CG (A) | 212 Describing GD for a periodic A-nontrivial near abelian group | 216 The algebraic commutator group of a periodic near abelian group | 217

10.1 10.2 10.3

11

A list of examples | 219

Part III: Applications Introduction | 227 12 12.1 12.2 12.3 12.4

Classifying topologically quasihamiltonian groups | 229 Generalities | 229 The p-group case | 231 The periodic case | 233 The nonperiodic case | 235

13 Locally compact groups with a modular subgroup lattice | 241 13.1 Generalities | 241 13.2 The p-group case | 245 13.2.1 Abelian p-groups | 245 13.2.2 Nonperiodic locally compact abelian groups | 249 13.3 Iwasawa (p, q)-factors | 249 13.4 The periodic case | 252 13.5 A summary of periodic locally compact topological M-groups | 257 13.6 The nonperiodic case | 259 14 Strongly topologically quasihamiltonian groups | 263 14.1 Historical background | 263 14.2 More notation | 264 14.3 Generalities | 264 14.4 The abelian case | 265 14.4.1 The Abelian p-group case | 266

XIV | Contents 14.4.2 14.5 14.5.1 14.5.2

Classifying periodic abelian groups | 271 The nonabelian situation | 274 The periodic case | 274 The nonperiodic case | 283

Bibliography | 289 List of symbols | 295 Index | 297

Overview This treatise aims at familiarizing readers with a class of locally compact groups which will be called near abelian. Inevitably, many aspects of our approach are of a technical nature and will involve our inspecting general techniques applying to locally compact groups quite generally. We therefore believe that we serve our readers best by presenting a gentle overview of the scope of the book and its principal results and achievements. We are dealing with locally compact groups. The very words mean different things to different people. We are therefore not amiss when we begin right away with a historical review of the genesis of locally compact groups.

Glancing back: How do locally compact groups arise? The history of the structure theory of locally compact groups is a long one. One of its roots is David Hilbert’s question of 1900 whether a locally euclidean topological group might possibly support the introduction of a differentiable parametrization such that the group operations are in fact differentiable. A first affirmative answer was given for compact locally euclidean groups when they were found to be real matrix groups as a consequence of the foundational work by Hermann Weyl and his student Fritz Peter in 1927 on the representation theory of compact groups. A second step was achieved when the answer was found to be “yes” for commutative locally compact groups. This emerged out of the fundamental duality theorems by Lev Semyonovich Pontryagin (1934) and Egbert van Kampen (1937). This duality forever determined the structure and harmonic analysis of locally compact abelian groups after it was widely read in the book of 1940, 1953, 1965, and 1979 by André Weil on the integration in locally compact groups [110]. A final positive answer to Hilbert’s Fifth Problem had to wait almost another two decades until, 17 years after the Second World War, the contributions of Andrew Mattei Gleason, Deane Montgomery, and Leo Zippin around 1952 provided the final affirmative answer to Hilbert’s problem. It led almost at once to the fundamental insights of Hidehiko Yamabe 1953, completing the pioneering work of Kenkichi Iwasawa (1949) and providing the fundamental structure of all those locally compact groups G which had a compact space G/G0 of connected components: Such groups were recognized as being approximated by quotient groups G/N modulo arbitrarily small compact normal subgroups N in such a fashion that each G/N is a Lie group, that is, one of those groups on which Hilbert had focussed in the fifth of his 23 influential problems in 1900 and which Sophus Marius Lie (1842–1899) had invented together with an ingenious algebraization method, long known nowadays under the name of Lie algebra theory. (See https://doi.org/10.1515/9783110599190-202

XVI | Overview [75].) A special case arises when all G/N are discrete finite groups; in this case G is called profinite. The solution of Hilbert’s Fifth Problem in the middle of the last century opened up the access to the structure theory of locally compact groups to the extent they could be approximated by Lie groups, due to the rich Lie theory meanwhile developed in algebra, geometry, and functional analysis. Recently, interest in Hilbert’s Fifth Problem was rekindled in the present century under the influence of Terence Tao [105]. There are nonstandard approaches to dealing with Hilbert’s Fifth Problem by Joram Hirschfeld in [49] and recently by Lou van den Dries and Isaac Goldbring in [106]. The quest for a solution to Hilbert’s Fifth Problem, at any rate, led to one major direction in the research of topological groups: In focus was the class of groups G approximable by Lie group quotients G/N, and finally G itself needed no longer to be locally compact. Such groups were called pro-Lie groups and considered for the sake of their own. (See [52] and [53].) Their theory reached as far as almost connected locally compact groups go (that is, those for which the group of its connected components G/G0 is compact, which we mentioned earlier), including all compact and all connected locally compact groups. But it did not truly reach further. Still, every locally compact group (and indeed every pro-Lie group) G has canonically and functorially attached to it a (frequently infinite-dimensional) Lie algebra L(G). The underlying topological vector space is of the form ℝJ for some set J (cf. [52]). Therefore the cardinal card(J) is canonically associated with G and one can argue that it well reflects its topological dimension DIM G (cf., e. g., [54, Scholium 9.54, p. 498 ff]). Indeed, we have the following information right away. Proposition 1. Let G be a topological group which is locally compact or is a pro-Lie group. Then the following statements are equivalent: (1) every connected component of G is singleton, that is, G is totally disconnected, (2) DIM G = 0, that is, G is zero-dimensional. If G is compact, then it is zero-dimensional if and only if it is profinite. For this book it is important to see that, in the course of the twentieth century, there was a second trend in the study of locally compact groups that is equally significant even though it is opposite to the concept of connectivity in topological groups. This trend is represented by the class of compact or locally compact zero-dimensional groups. Such groups were encountered in field theory at an early stage. Indeed, in Galois theory the consideration of the appropriate infinite ascending family of finite Galois extensions and, finally, its union would, dually, lead to an inverse family of finite Galois groups and, in the end, to their projective limit. Thus was produced what became known as a profinite group. The Galois group of the infinite field extension, equipped with the Krull topology, is thus a profinite group. One recognized soon that profinite

Glancing back: How do locally compact groups arise?

| XVII

groups and compact totally disconnected groups were one and the same mathematical object, expressed algebraically on the one hand and topologically on the other. Comprehensive literature on this class of groups appeared much later than textbooks on topological groups in which connected components played a leading role. The first monograph dealing explicitly with the subject of profinite groups appears to be the one by Stephen S. Shatz and just before the end of the twentieth century totally disconnected compact groups were the protagonists of books simultaneously entitled “Profinite Groups” by John Stuart Wilson and Luis Ribes jointly with Pavel Zalesskii in 1998, while George Willis in 1994 laid the foundations of a general structure theory of totally disconnected locally compact groups if no additional algebraic information about them is available (see [112]). On the other hand, in the realm of locally compact abelian groups, the completion of the field ℚ of rational numbers with respect to any nonarchimedian valuation yields the locally compact p-adic fields ℚp as a totally disconnected counterpart of the connected field ℝ of real numbers. The fields ℚp and their integral subrings ℤp were basic building blocks of ever so many totally disconnected groups, in particular the linear groups over these field and indeed all p-adic Lie groups which Nicolas Bourbaki judiciously included in his comprehensive treatise on Lie groups. Bourbaki’s text on Lie groups formed the culmination and certainly the endpoint of his encyclopedic project extending over several decades. The world of totally disconnected locally compact groups developed its own existence, methods, and philosophies, partly deriving from finite group theory via approximation through the formation of projective limits, partly through graph theory where the appropriate automorphism theory provides the fitting representation theory, and also through the general impact of algebraic number theory which in the textbook literature is indicated by the books of Helmut Hasse since the thirties (see, e. g., [41, 40]) and lastly, 1967, by the “Basic Number Theory” of André Weil (see [111]), whose book on the integration of locally compact groups of 1940 (with later editions through four decades) had influenced the progress of harmonic analysis of locally compact groups so much (see [110]). Significant new directions in the pursuit of totally disconnected locally compact groups are assembled in a collection edited by Caprace and Monod in [14]. This source is indicative of some important trends pursued in this area. We pointed out that each locally compact group G, irrespective of any structural assumption, has attached to it a (topological) Lie algebra L(G) (and therefore a universal cardinal-valued dimension). The more recent interest in zero-dimensional locally compact groups G has led to a new focus on another functorially attached invariant, namely, a compact Hausdorff space 𝒮𝒰ℬ(G) consisting of all closed subgroups of G endowed with a suitable topology and now frequently called the Chabauty space of G. This tool is not exactly new, but has been widely utilized in applications recently. The Chabauty space is a special case of what has been called the hyperspace of a compact (or locally compact) space first introduced by Vietoris (see [31]). In topological algebra

XVIII | Overview hyperspaces were used and described, e. g., in [8, 55, 56], and all recent publications where the name of Chabauty appears in the title (e. g., [14, 37, 39, 38], and [22]).

Introductory definitions and results We now address the content of this book and therefore return to the second thrust of the study of locally compact groups, which is concerned with the research of zerodimensional groups. Definition 2. A topological group G is called periodic if (i) G is locally compact and totally disconnected and (ii) ⟨g⟩ is compact for all g ∈ G. So a compact group is periodic if and only if it is profinite. A very significant portion of the locally compact groups considered in this treatise will be periodic groups. That is, we shall deal with totally disconnected locally compact groups in which every element is contained in a profinite subgroup. More specifically, we shall say that a topological group G is compactly ruled if it is the directed union of its compact open subgroups. If G is a locally compact solvable group in which every element is contained in a compact subgroup, then it is compactly ruled. (See Remark 1.14.) The class of compactly ruled groups comprises both the class of profinite groups and the one of locally finite groups, i. e., groups where every finite subset generates a finite subgroup only; see the book of O. H. Kegel and B. A. F. Wehrfritz [64] for a comprehensive study of this class of groups. In many cases we shall assume that the periodic groups we will consider are compactly ruled. These properties make them topologically special; just how close they make our groups to profinite groups remains to be seen in the course of the book. A second significant property of the groups we study is an algebraic one: they are solvable, indeed metabelian. Again, it is another challenge to discern just how close this makes them to abelian groups. The groups we study will be called near abelian. In order to offer a precise definition of this class of locally compact groups we need one preliminary definition, extending a very familiar concept. Definition 3. A topological group G will be called monothetic if G = ⟨g⟩ for some g ∈ G and inductively monothetic if for every finite subset F ⊆ G there is an element g ∈ G such that ⟨F⟩ = ⟨g⟩. Remark 4. In the literature, the discrete inductively monothetic groups have been termed locally cyclic; see, e. g., [32] and [64]. We shall discuss and classify inductively monothetic locally compact groups in greater detail later; but let us observe here right away a connected example illustrating the two definitions: Indeed let 𝕋 = ℝ/ℤ denote the (additively written) circle group.

Some history of near abelian groups | XIX

Then the 2-torus is monothetic but is not inductively monothetic, since ( 21 ⋅ℤ/ℤ)2 ⊆ 𝕋2 is finitely generated but is not monothetic. Yet in the domain of totally disconnected locally compact groups, a monothetic group is either ℤ or the character group of a subgroup of the discrete group ℚ/ℤ and thus is a product of procyclic p-groups as we shall review below in Proposition 10. Our classification Theorem 12 of inductively monothetic groups then yields that every zero-dimensional monothetic group is inductively monothetic. In a periodic group, each monothetic subgroup ⟨g⟩ is compact, equivalently, procyclic, i. e., the projective limit of an inverse system of finite cyclic groups. Now we are prepared for a definition of the class of locally compact groups whose details we shall consider here (see Definition 6.1). Definition 5. A topological group G is near abelian provided it is locally compact and contains a closed normal abelian subgroup A such that 1. G/A is an abelian inductively monothetic group and 2. every closed subgroup of A is normal in G. The subgroup A we shall call a base or a base group for G. A given near abelian group G may have many base groups as is illustrated by the example of G = ℤ and A any subgroup. When we eventually collect applications for this class of locally compact groups, then we shall see that for instance all locally compact groups in which two subgroups commute setwise form a subclass of the class of near abelian groups and that the class of all locally compact groups in which the lattice of closed subgroups is modular is likewise a subclass of the class of near abelian groups. (See Part III.)

Some history of near abelian groups In the world of discrete groups, near abelian groups historically appeared in a natural way when K. Iwasawa attempted the classification of what is now known as quasihamiltonian and modular groups as expounded in the monograph by R. Schmidt [97]. It was F. Kümmich (cf. [69]) who initiated in his dissertation written under the direction of Peter Plaumann and in papers developed from his thesis the study of topologically quasihamiltonian groups. These are topological groups such that XY = YX is valid for any closed subgroups X and Y of such a group. Yu. Mukhin, who in [78] dealt with locally compact abelian topologically modular groups, turned to investigating the nonabelian situation in [81]. The properties which we consider as the defining ones, namely, that there be a closed normal abelian subgroup A of G such that G/A is inductively monothetic and such that every closed subgroup of A is normal in G, suggest themselves by the fact, proved by K. Iwasawa, that discrete quasihamiltonian groups satisfy them. In a similar

XX | Overview vein, Mukhin, during his work on classifying topologically modular groups, finds that his groups, under additional assumptions, are all near abelian in our sense (see, e. g., [81]). A more recent article, by K. H. Hofmann and F. G. Russo, was devoted to classifying compact p-groups that are topologically quasihamiltonian (cf. [57]) and introduced the concept of near abelian groups in their more special context for the first time, in which inductively monothetic groups were simply p-procyclic groups. The major result states that such groups are at the same time topologically quasihamiltonian group and near abelian with the exception of p = 2, in which case some sporadic near abelian groups are topologically quasihamiltonian while the bulk of them are not. This once again is evidence of the fact that is often quoted by number theorists and group theorists alike that 2 is the oddest of all primes. In linear algebra, a group G of (n + 1) × (n + 1)-matrices of the form r⋅E ( n 0

v ), 1

0 < r ∈ ℝ, v ∈ ℝn

with the identity En of GL(n, ℝ) is a metabelian Lie group that has been called almost abelian (see [48], p. 408, Example V.4.13). The subgroup A of all matrices with r = 1 is isomorphic to ℝn and every vector subspace of A is normal in G and G/A ≅ ℝ is a one-dimensional Lie group which is not inductively monothetic, but we shall see that inductively monothetic groups are in some sense “rank one” group analogs. In both cases we have a representation ψ: G/A → Aut(A),

ψ(gA)(a) = gag −1

as an essential element of the structure. In the near abelian case we shall say that G is A-nontrivial if the image of ψ has more than two elements. Whereas in the Lie group case, the structure of an almost abelian Lie group G is comparatively simple, in the case of a group G satisfying the conditions of Definition 5 it is likely to be rather sophisticated as we illustrate by the following result (see Theorem 7.11) in which CG (A) = ker ψ denotes the centralizer {g ∈ G : (∀a ∈ A) ag = ga} of A in G. Theorem 6 (Structure theorem I on near abelian groups). Let G be an A-nontrivial near abelian group. Then (1) A is periodic, (2) G is totally disconnected, (3) when ψ(G/A) is compact or A is an open subgroup, then G has arbitrarily small compact open normal subgroups, that is, G is prodiscrete, (4) G itself is periodic if and only if G/A is periodic if and only if G/A is not isomorphic to a subgroup of the discrete group ℚ of rational numbers, (5) CG (A) is an abelian normal subgroup containing A and is maximal for this property.

Inductively monothetic groups | XXI

This shows that for the topic of this book, periodic locally compact groups will play a significant role.

Inductively monothetic groups A good understanding of near abelian groups depends on a clear insight into the concept of inductively monothetic groups. They were also recently featured in [38]. Here we must recall the concept of a local product of a family of topological groups which, in the theory of locally compact groups, mediates between the idea of a Tychonoff product of compact groups and the idea of a direct sum of a family of discrete groups; the principal applications are in the domain of abelian groups, but the concept as such has nothing to do with commutativity. Definition 7. Let (Gj )j∈J be a family of locally compact groups and assume that for each j ∈ J the group Gj contains a compact open subgroup Cj . Let P be the subgroup of the cartesian product of the Gj containing exactly those J-tuples (gj )j∈J of elements gj ∈ Gj for which the set {j ∈ J : gj ∉ Cj } is finite. Then P contains the cartesian product C := ∏j∈J Cj which is a compact topological group with respect to the Tychonoff topology. The group P has a unique group topology with respect to which C is an open subgroup. Now the local product of the family ((Gj , Cj ))j∈J is the group P with this topology, and it is denoted by loc

P = ∏(Gj , Cj ). j∈J

(See Definition 2.49.) Let us note that the local product is a locally compact group with the compact open subgroup ∏i∈J Cp . While the full product ∏j∈J Gj has its own product topology we note that the local product topology on P in general is properly finer than the subgroup topology. The concept of the local product was introduced and its duality theory in the commutative situation was studied by J. Braconnier in [12]. For us local products play a role that most frequently appear with J being the set π of all prime numbers. This is well illustrated by the following key result on periodic locally compact abelian groups, where we note that for a locally compact abelian group G and each prime p, we have a unique characteristic subgroup Gp containing all elements g for which ⟨g⟩ is a profinite p-group, i. e., a p-procyclic subgroup; Gp is called the p-primary component or the p-Sylow subgroup of G. Theorem 8 (J. Braconnier). Let G be a periodic locally compact abelian group and C any compact open subgroup of G. Then G is isomorphic to the local product loc

∏ (Gp , Cp ).

p∈π

XXII | Overview The following remark is useful for us as a consequence of the facts that any compact p-group Cp is a ℤp -module and that any prime q ≠ p is a unit in ℤp ; accordingly, Cp is divisible by n ∈ ℕ with (n, p) = 1. Remark 9. A periodic locally compact abelian group G is divisible iff all p-Sylow subgroups Gp are divisible. The structure of a locally compact monothetic group G is familiar to workers in the area: It is either isomorphic to the discrete group ℤ of integers or is compact (Weil’s Lemma; see, e. g., [54], Proposition 7.43, p. 348.). A compact abelian group is known if its discrete Pontryagin dual is known. A compact abelian group G is monothetic if and only if there is a morphism f : ℤ → G of locally compact groups with dense image, that ̂ into the character group 𝕋, that is, iff is, iff there is an injection of the discrete group G (2ℵ0 ) ̂ ⊕ ⨁p∈π ℤ(p∞ ). (Here ℤ(p∞ ), as usual, is the G is isomorphic to a subgroup of ℚ 1 Prüfer’s p-group ⋃n∈ℕ pn ℤ/ℤ ⊆ 𝕋.) Whenever G is zero-dimensional, things simplify dramatically. Proposition 10. A compact zero-dimensional abelian group G is monothetic iff it is isomorphic to ∏p∈π Gp where the p-factor Gp is either ℤ(pm ) for some m ∈ ℕ0 = {0, 1, 2, . . . } or ℤp , the additive group of the ring of p-adic numbers. In the area of compact groups, zero-dimensional groups are often called profinite and monothetic groups are also known as procyclic. Let us now proceed to inductively monothetic locally compact groups. For periodic inductively monothetic groups it is convenient to introduce some special terminology. From Braconnier’s Theorem 8 we know that every periodic locally compact abelian group G, for any given compact open subgroup C ⊆ G, is (isomorphic to) the local product loc

∏ (Gp , Cp )

p∈π

of its p-Sylow subgroups Gp . Definition 11. A topological group G is called Π-procyclic, if it is a periodic locally compact abelian group and each p-Sylow subgroup Gp is either a finite cyclic p-group (possibly singleton) or ℤp , that is, Gp is p-procyclic. Now we can formulate our classification of inductively monothetic locally compact groups. Theorem 12 (Classification theorem of inductively monothetic groups). Let G be an inductively monothetic locally compact group. Then G is either (a) a one-dimensional compact connected abelian group, (b) a subgroup of the discrete group ℚ, or

Inductively monothetic groups | XXIII

(c) a periodic locally compact abelian group such that Gp is isomorphic to ℚp , or ℤ(p∞ ), or ℤp , or ℤ(pnp ) for some np ∈ ℕ0 = {0, 1, 2, . . . }. (See Theorem 5.15.) All inductively monothetic groups are sigma-compact, i. e., are countable unions of compact subsets. The connected groups in Theorem 12 are monothetic; other types may or may not be monothetic. The periodic inductively monothetic groups G of part (b) require special attention. First we divide the set π of all prime numbers into disjoint sets, i. e., – πA = {p ∈ π : Gp ≅ ℚp }, – πB = {p ∈ π : Gp ≅ ℤ(p∞ )}, – πC = {p ∈ π : Gp ≅ ℤp }, – πD = {p ∈ π : (∃n ∈ ℕ0 ) Gp ≅ ℤ(pn )}. Now we fix a compact open subgroup C of G and identify G with the local product ∏p∈π (Gp , Cp ). Further, we define two closed characteristic subgroups as D := P :=

loc

∏ (Gp , Cp ),

p∈πA ∪πB loc

∏ (Gp , Cp ),

p∈πC ∪πD

and we notice that G = D ⊕ P. Both subgroups D and G are characteristic, and we notice that in view of Remark 9 D is the unique largest divisible subgroup of G and P is reduced. Theorem 13 (Classification of inductively monothetic groups, continued). Let G be a periodic inductively monothetic locally compact group. Then G is the direct topological and algebraic sum D ⊕ P of two characteristic closed subgroups of which D is the largest divisible subgroup of G and P is the unique largest Π-procyclic subgroup according to Definition 11. We apply this information to the structure theory of a near abelian periodic group G with base A. Then G/A = D ⊕ P as in Theorem 13. Lemma 14. Let GD and GP denote the full inverse images of D and P, respectively, for the quotient morphism G → G/A. Now GD and GP are closed normal subgroups of G such that G = GD GP and GD ∩ GP = A; moreover, GD ⊆ CG (A). (See Theorem 7.14.) Theorem 15 (Structure theorem II on near abelian groups). Let G be a periodic near abelian locally compact group with a base A such that G is A-nontrivial. Then A ⊆

XXIV | Overview GD ⊆ CG (A), where GD is a characteristic abelian subgroup such that G/GD ≅ GP /A is Π-procyclic. (See Theorem 7.11.) This portion of the basic structure theory of near abelian groups in the periodic situation will allow us to concentrate largely on the case that the factor group G/A is Π-procyclic.

Factorization and scaling We begin with a definition elaborating the definition of near abelian groups in greater detail. Definition 16. Let G be a near abelian locally compact group with a base A. A closed subgroup H is called a scaling subgroup for A if (i) H is inductively monothetic and (ii) G = AH. Example 17. There exists a (discrete) abelian group G with a subgroup A which is not a direct summand and which has the following properties: A is the torsion subgroup 1 of G of the form A ≅ ⨁n∈ℕ ℤ(p) and G/A ≅ ⋃n∈ℕ 2⋅3⋅⋅⋅p ℤ ⊆ ℚ. n

Example 18. Prüfer’s example from [32, p. 150] is a group G ≅ ⨁n∈ℕ ℤ(pn ) and it contains a subgroup A which is not a direct summand such that G/A ≅ ℤ(p∞ ). (See Example 7.42.) In Example 17, the group G is a subgroup of ℝ/ℤ and is a construction due to D. Maier [73] and [74]. Also Example ∇ in Theorem A1.32, p. 686 of [54] displays such nonsplitting phenomena. These examples show that there are obstructions to a very general result asserting the existence of a scaling group for a near abelian group G with a base A. A scaling group H, whenever it exists, is a supplement for A in G but not in general a semidirect complement. How far a supplement is from being a complement can be clarified under fairly general circumstances; we illustrate that in the following proposition (see Proposition 1.31). Proposition 19. Let G be a locally compact group with a closed normal subgroup A and a closed sigma-compact subgroup H containing a compact open subgroup and satisfying G = AH. The inner automorphisms define a morphism α: H → Aut(A) by α(h)(a) = hah−1 . Then we have the following conclusions:

Factorization and scaling

| XXV

(i) The semidirect product A ⋊α H is a locally compact group and the function μ: A ⋊α H → G, μ(a, h) = ah, is a quotient morphism with kernel {(h−1 , h) : h ∈ A ∩ H} isomorphic to A ∩ H, mapping both A and H faithfully. (ii) The factor group G/(A ∩ H) is a semidirect product of A/(A ∩ H) and H/(A ∩ H) and the composition A ⋊α H → G → G/(A ∩ H) is equivalent to the natural quotient morphism A ⋊α H →

A H ⋊ A∩H A∩H

with kernel (A ∩ H) × (A ∩ H). Notice that a scaling subgroup H of a near abelian group is sigma-compact and has a compact open subgroup, so that the proposition applies in its entirety to near abelian locally compact groups. The typical “sandwich situation” A ⋊ H → AH →

A H ⋊ A∩H A∩H

is also observed in significant ways in the structure theory of compact groups (see [54], e. g., Corollary 6.75 ff.). So one of the most pressing questions of the structure theory of near abelian locally compact groups is the following. Problem 1. Under which conditions does a locally compact group G with a normal subgroup A such that G/A is inductively monothetic contain a closed inductively monothetic subgroup H such that G = AH? If G/A is in fact monothetic, then the answer is affirmative provided G is periodic. A more detailed answer will be given in Proposition 5.33. Under more general circumstances we shall prove in this book the following theorems giving a partial answer to Problem 1 (see Theorems 5.41 and 1.33). Theorem 20. Let G be a locally compact group with a compact normal subgroup A such that G/A is Π-procyclic. Then G contains a Π-procyclic subgroup HΠ such that G = AHΠ . Theorem 21. Let G be a locally compact group with a compact open normal subgroup A such that G/A is isomorphic to an infinite subgroup of the group ℚ. Then G contains a discrete subgroup H ≅ G/A such that G is a semidirect product AH ≅ A ⋊ H. It would be highly desirable to have such theorems without the hypothesis that A be compact. The proofs of these theorems (see Theorem 5.32 and the subsequent discussion) make essential use of the compact Hausdorff Vietoris–Chabauty space 𝒮𝒰ℬ(G) which is attached to every locally compact group as a general invariant.

XXVI | Overview As long as this approach requires the compactness of A, the following theorem may be considered as fundamental for the structure theory of near abelian locally compact groups because for certain near abelian groups we obtain scaling groups without the hypothesis of the compactness of A. Theorem 22. Let G be a locally compact near abelian group with a base A such that G is A-nontrivial and G/A is Π-procyclic. Then G contains a Π-procyclic scaling subgroup HΠ for A with G = AHΠ . For a proof see Theorem 7.36. The proof requires a wide spectrum of parts of our general structure theory of near abelian groups discussed in this book. In particular, at the root of this existence theorem is the Chabauty space 𝒮𝒰ℬ(G) of the group G which we mentioned earlier. This theorem and Theorem 15 now yield the following theorem (see Theorem 7.41). Theorem 23. Any periodic locally compact near abelian group G with base A for which G is nontrivial possesses a Π-procyclic closed subgroup HΠ such that G = GD HΠ for the abelian normal subgroup GD with A ⊆ GD ⊆ CG (A). In particular, Proposition 19 then shows us that we have the following. Corollary 24. Any periodic locally compact near abelian group G is a quotient of GD ⋊HΠ modulo a subgroup isomorphic to GD ∩ HΠ . The center of a group G will be denoted by Z(G). Corollary 25. For every periodic locally compact near abelian group G with base A for which G is nontrivial and for which G/A is Π-procyclic, we have CG (A) = AZ(G) and CG (A) ∩ HΠ ⊆ Z(G), that is, AZ(G) ∩ HΠ = Z(G) ∩ HΠ . (See Theorem 6.4, parts (iv) and (v) and their proofs in combination with Theorem 7.36.) The following theorem then is rather definitive on the factorization of a periodic near abelian locally compact group and may be considered as one of the main theorems on their structure. We recall the natural decomposition G = GD GP , GD ∩ GP = A from Theorem 15. Theorem 26 (Structure theorem III on periodic near abelian groups). Let G be a locally compact near abelian group G with a base A such that G is not A-trivial. Then (i) GP has a Π-procyclic scaling group H for A so that GP = AH, (ii) CG (A) = GD Z(GP ) and CGP (A) = AZ(GP ), (iii) G = GD Z(GP )H, (iv) AZ(GP ) ∩ H = Z(GP ) ∩ H. (See Theorem 7.43.)

The Sylow theory of periodic groups | XXVII

Example 18 shows that GD is not prone to decompose as a product AK where K is an inductively monothetic scaling group such that K/(K ∩ A) ≅ D is a divisible inductively monothetic group. In this context the following is a largely untreated topic. Problem 2. Clarify the structure of a periodic locally compact abelian group G with a closed subgroup A such that G/A is inductively monothetic and divisible. According to Braconnier’s local product decomposition theorem for locally compact abelian periodic groups, this problem reduces to locally compact periodic p-groups for a prime number p and thus, in particular, to the case that G/A is isomorphic either to ℚp or to the discrete group ℤ(p∞ ) and that A is a locally compact periodic p-group. For general information on near abelian locally compact groups G with base groups A concerning closed subgroups A∗ ⊆ AZ(GP ) containing A may still be taken as base subgroups of G; see Theorem 10.18. The role of the center Z(GP ) in CG (A) = AZ(GP )—a locally compact abelian group we know to be a local product of its p-primary components Ap Z(GP )p —is still a bit mysterious; more information will be forthcoming in Theorem 62 below and Proposition 10.29.

The Sylow theory of periodic groups A Sylow theory, i. e., a discussion of the existence and conjugacy of maximal p-subgroups, and, more generally, of maximal σ-subgroups where σ is a set of primes, is available for profinite groups (see [93, 98, 113]). Several attempts have been made in order to generalize Sylow theory to noncompact and locally compact groups; see, e. g., the survey from 1964 by Čarin, [17], the one by Platonov in [91] or, more recently, by Reid in [92]. Here we shall focus on our class of compactly ruled groups. If the topology on a compactly ruled group is discrete, the group is locally finite, i. e., every finite subset generates a finite subgroup. Then our Sylow theory reduces to the one presented in the book of Kegel and Wehrfritz; see [64]. In each locally compact periodic group G the concept of a p-group can be defined meaningfully. Indeed if g ∈ G, then M := ⟨g⟩ is a zero-dimensional monothetic compact group, and thus M ≅ ∏ Mp , p∈π

where for each prime p the p-primary component Mp is either ≅ ℤp or ℤ(pnp ) for some np = 0, 1, 2, . . . . It is practical to generalize the concept of a p-element: For each subset σ ⊆ π, an element g ∈ G is called a σ-element if ⟨g⟩ = ∏p∈σ Mp ; if σ = {p}, then g is called a p-element. The group G is a σ-group, if all of its elements are σ-elements. A subgroup S

XXVIII | Overview is called a σ-Sylow subgroup of G if it is a maximal element in the set of σ-subgroups. A simple application of Zorn’s lemma shows that every σ-element is contained in a σ-Sylow subgroup. We record the following. Lemma 27 (The closure lemma). Let G be any locally compact totally disconnected group. Then for any subset σ ⊆ π, the set Gσ of all σ-elements of G is closed in G. (See Lemma 2.6.) Let us now look at some traditional splitting theorems that we shall show to work in the general background of periodic locally compact groups.

The Schur–Zassenhaus splitting The splitting of finite groups into products of subgroups of relatively prime orders can be generalized to the locally compact setting up to a point. For locally finite groups the results to be discussed are well known; see, e. g., [64]. They also relate to work of the second author; see [51, 55]. Proposition 28. Let N be a closed subgroup of a locally compact periodic group G and assume N ⊆ Gσ . Then the following conditions are equivalent: (1) N is a normal Sylow subgroup, (2) N = Gσ , (3) N is normal and G/N contains no p-element with p ∈ σ. (See Proposition 2.18.) Definition 29. Let G be a locally compact periodic group and N a closed subgroup. We say that N satisfies the Schur–Zassenhaus condition if and only if it satisfies the equivalent conditions of Proposition 28 above for σ = π(N). Theorem 30 (Schur–Zassenhaus theorem). Let G be a periodic group and N a closed subgroup satisfying the following two conditions: (1) N satisfies the Schur–Zassenhaus condition, (2) G/N is a directed countable union of compact subgroups. Then the following conclusions hold: (i) N possesses a complement H in G, (ii) let K be a closed subgroup of G such that K ∩ N = {1} and assume that G/N is compact; then there is a g ∈ G such that gKg −1 = H. (See Theorem 2.21.)

Sylow subgroups commuting pairwise

| XXIX

It should be remarked that for solvable groups (such as near abelian groups) the periodic groups are always directed unions of their open compact subgroups. In such a situation condition (2) simply means that G/N is sigma-compact. The Schur–Zassenhaus configuration in the locally compact environment is delicate, since problems do arise with the product of a closed normal subgroup and a closed subgroup; such a product need not be closed, in general. Still we do have theorems like the following. Theorem 31. Let N be a normal σ-Sylow subgroup of a locally compact periodic group G. Then a (π \ σ)-Sylow subgroup H of G exists such that NH is an open and hence closed subgroup. Moreover, if H is any (π \ σ)-Sylow subgroup of G, then NH is closed in G and H is a complement of N in NH, that is, NH = N ⋊ H. (See Theorem 2.25.)

Sylow subgroups commuting pairwise Let p ∈ π denote any prime and p󸀠 := π \ {p}. Then we have the following result. Lemma 32. For a compactly ruled group G and a prime number p, the following conditions are equivalent: (1) [Gp , Gp󸀠 ] = {1}, (2) Both Gp and Gp󸀠 are subgroups, and G = Gp × Gp󸀠 , (3) There is a unique projection prp : G → Gp with kernel Gp󸀠 . (See Theorem 2.44.) Definition 33. For a periodic locally compact group G we write ν(G) = {p ∈ π : [Gp , Gp󸀠 ] = {1}}. We have found the following structure theorem very useful in the context of near abelian generalizing the well-known fact that a pronilpotent group is the cartesian product of its Sylow subgroups (cf. [93]). Theorem 34. In a compactly ruled locally compact group G, the set Gν(G)󸀠 of α-elements with α ∩ ν(G) = 0 is a closed normal subgroup, and all p-Sylow subgroups for p ∈ ν(G) are normal subgroups. Moreover, loc

G ≅ Gν(G)󸀠 × ∏ (Gp , Up ) p∈ν(G)

for a suitable family of compact open subgroups Up ⊆ Gp as p ranges through ν(G). (See Theorem 2.53.)

XXX | Overview

The internal structure of Sylow subgroups of near abelian groups For periodic near abelian groups, to which we can apply a Sylow theory meaningfully, we assume that G is a periodic near abelian locally compact group such that G is nontrivial for a base A. Theorem 35. Let G be a periodic near abelian group and A a base for which G is A-nontrivial and which satisfies A = CG (A). Then, for every set σ of prime numbers, there is a σ-Sylow subgroup Sσ . Fix a σ-Sylow subgroup Sσ of G. Then (i) Sσ ∩ A is the σ-primary component Aσ of A, or equivalently, the σ-Sylow subgroup of A, (ii) Sσ /Aσ ≅ Sσ A/A = (G/A)σ , (iii) if (G/CG (A))σ ≅ H/(H ∩ CG (A)) is compact, then any two σ-Sylow subgroups of G are conjugate, (iv) Sσ = CG (A)σ Hσ = Aσ Z(Gp )σ Hσ , where H is as in Theorem 23. (See Theorem 7.15.) The case that σ = {p} is an important special case. Let us note that it may happen that Sp ⊆ A, in which case we have Hp = {1}.

On the structure of locally compact abelian groups The basic subject of this book is the structure of periodic groups in a class of metabelian locally compact groups. It is to be expected that, at one point sooner or later, we have to come back to the structure of locally compact abelian groups itself. It may be somewhat surprising that in an area so well understood as locally compact abelian groups we would have to unearth information relevant to a new topic. The reason is the following. While structural information on a compact abelian group G is fully transportable ̂ for a noncompact to purely algebraic information on its discrete character group G, locally compact abelian group G, while all information is still reflected completely in ̂ in this case G ̂ is again a nondiscrete locally compact group. In its character group G, a certain sense, therefore, the passage to the character group is circular in the locally compact case. This is evidenced in Jean Braconnier’s pioneering paper [12] which we have cited before, in Theorem 8, where we introduced his concept of a local product. Indeed, if a locally compact abelian group is a local product G = ∏loc j∈J (Gj , Cj ) of locally compact abelian groups Gj with G having the compact open subgroup C = ∏j∈J Cj , then ̂, C ⊥ ), where C ⊥ denotes the annihilator of C ⊆ G in ̂ ≅ ∏loc (G its character group is G j j j j j j∈J ̂. This dualization we shall use in Lemma 3.82. (For the annihilator mechanism see G j

[54], Lemma 7.17 ff.) At any rate, information on noncompact locally compact abelian groups is ample in the literature, but not everything we need is readily at hand.

On the structure of locally compact abelian groups | XXXI

In dealing with locally compact abelian groups, one fundamental background structure theorem is the so-called “vector space splitting theorem” (see [54], Theorem 7.57, p. 354). Theorem 36. (i) Every locally compact abelian group G is algebraically and topologically of the form G = E ⊕ H for a subgroup E ≅ ℝn and a locally compact abelian subgroup H such that the following conditions hold: (a) H contains a compact subgroup C which is open in H, (b) H contains the group comp(G) of all elements contained in some compact subgroup, (c) H0 = (comp(G))0 = comp(G0 ) is the unique maximal compact connected subgroup of G, (d) The open closed fully characteristic subgroup G1 = G0 + comp(G) of G is isomorphic to E ⊕ comp(G), (e) G/G1 is a discrete torsion-free group and G1 is the smallest open subgroup with this property. (ii) The group G is periodic if and only if (a) G0 = E ⊕ H0 = {0} and (b) G = comp(G). (iii) G0 = E ⊕ H0 is divisible. G G1 G0 E

H K

K = comp(G), G1 =G0 +K=E ⊕ K.

K0 0

The vector space splitting theorem above and Braconnier’s primary decomposition Theorem 8 serve two purposes at this point. Firstly, they give an impression where periodic locally compact abelian groups are positioned in the universe of locally compact abelian groups. Secondly, they convince us that the investigation of the structure of periodic locally compact abelian groups means the investigation of the structure of locally compact abelian p-groups. And thirdly, since the p-adic completion ℚp of ℚ in many respects is an analog of the archimedian completion ℝ of ℚ, they raise the

XXXII | Overview question whether there is an analog of the vector space splitting theorem in the world of locally compact abelian p-groups, and indeed what divisible groups look like in locally compact abelian p-groups. We shall give some pertinent answers in this text. A very striking and instructive example and some of the consequences which its existence implies are described in Section 3.3.1 and notably in Theorem 3.17. Indeed let S denote the discrete torsion p-group ⨁n∈ℕ ℤ(pn ). We shall describe a reduced locally compact abelian p-group ∇p with a compact open subgroup ∇p󸀠 ≅ ℤp so that the following theorem is true. Theorem 37. There are two nonsplit exact sequences 0 → ℤp → ∇p → S → 0,

0 → S → ∇p → ℚp → 0.

(1) (2)

As a consequence of this theorem, the torsion subgroup tor ∇p is discrete and isomorphic to S, and we shall show that ∇p is isomorphic to a closed subgroup of ℚp × S. Let ϵ ∈ ∇p be the generator corresponding to 1 ∈ ℤp . Recall that an element g in an additively written group G is called divisible if the equation n⋅x = g is solvable in G for any n ∈ ℕ and that in a locally compact p-group this means that pn ⋅x = g is solvable for any n ∈ ℕ. Theorem 38. The element ϵ ∈ ∇p is divisible, and if G is any locally compact abelian p-group with a divisible element g then there is a continuous morphism ∇p → G mapping ϵ to g. ̂p is We observe right away that, due to Pontryagin duality, the character group ∇ a locally compact abelian p-group for which the following theorem holds with the compact group P = ∏n∈ℕ ℤ(pn ). Theorem 39. There are two nonsplit exact sequences ̂p → ℤ(p∞ ) → 0, 0→P→∇ ̂p → P → 0. 0 → ℚp → ∇

(1) (2)

In particular, this shows that ℚp can be isomorphic to a nonsplit closed subgroup with a compact quotient and thus shows a behavior that is radically different from that of ℝ as exhibited in the vector space splitting Theorem 36. Another interesting aspect of this discussion is the observation how far the group P is away from its torsion subgroup. We therefore record (see Corollary 3.20) the following. Proposition 40. The group P = ℤ(p)×ℤ(p2 )×ℤ(p3 )×⋅ ⋅ ⋅ contains a dense ℤp -submodule which is algebraically isomorphic to the ℤp -module ℤ(ℕ) p . Let us now look at locally compact abelian p-groups separated into the disjoint classes of torsion groups and torsion-free groups.

On the structure of locally compact abelian groups | XXXIII

Let us look at locally compact abelian torsion groups first. Any compact abelian torsion group is known to be a product of cyclic groups whose orders are bounded (and consequently are products of their finite set of p-Sylow subgroups; see, e. g., [54], Corollary 8.9 (iii).) We record the following fact (see Proposition 3.11). Proposition 41. Every locally compact abelian torsion p-group has a compact open subgroup of the form I

I

ℤ(p)I1 × ℤ(p2 ) 2 × ⋅ ⋅ ⋅ × ℤ(pn ) n for a finite collection of sets Ik , k = 1, . . . , n. We observe here that two compact open subgroups C1 and C2 of a topological group may differ, but they are commensurable, that is, the indices of C1 ∩ C2 in C1 , respectively, C2 are finite. (See Definition 3.2.) As a consequence of the two preceding propositions we conclude the following. Corollary 42. For any family (nj )j∈J of natural numbers, the profinite p-group G = ∏j∈J ℤ(pnj ) is either a torsion group or else it contains a ℤp -submodule isomorphic 2 3 to ℤ(ℕ) p whose closure is isomorphic to P = ℤ(p) × ℤ(p ) × ℤ(p ) × ⋅ ⋅ ⋅ . Each locally compact abelian torsion p-group G has a closed characteristic subgroup S(G) = {g ∈ G : p⋅g = 0} called its socle; obviously it has exponent p and therefore is a vector space over GF(p). The following is an immediate consequence of the preceding Proposition 41, and it applies to all p-socles of periodic locally compact abelian groups. Corollary 43. Any exponent p locally compact abelian group G is a direct product of a compact and a discrete exponent p-subgroup. Specifically, there are sets I1 and I2 such that G ≅ ℤ(p)(I1 ) × ℤ(p)I2 . Furthermore, if G is any locally compact torsion p-group, then rankp G = card I1 + card I2 . Every abelian group contains a unique largest divisible subgroup. By what we saw in Proposition 41 a compact abelian torsion group contains no divisible element other than the identity. The Prüfer group ℤ(p∞ ) is a discrete (hence locally compact) divisible torsion group. The question arises whether the largest divisible subgroup of a locally compact abelian torsion group is necessarily closed. The answer is negative as Example 3.28 shows. Indeed, if ℤ(p∞ ) = p1∞ ℤ/ℤ is a concrete representation of

Prüfer’s group and ℤ(p) = p1 ℤ/ℤ is the unique order p subgroup, then the local power

G = (ℤ(p∞ ), ℤ(p))loc, ℕ (cf. Definition 2.50) is a locally compact torsion group with

XXXIV | Overview compact open socle S= ℤ(p)ℕ and with a dense proper largest divisible subgroup ℤ(p∞ )(ℕ) whose socle is ℤ(p)(ℕ) . The group G we have just considered can be regarded as an open subgroup of the divisible group Δ = ℤ(p∞ )ℕ . In this fashion, Δ is defined to be a locally compact abelian divisible group. However, it is not a torsion group. Its torsion subgroup ∞

D := ⋃ ( n=0

ℕ

1 ℕ ℤ/ℤ) ⊆ ℤ(p∞ ) n p

is a locally compact abelian nondiscrete divisible torsion group, and we notice S ⊆ G ⊆ D ⊆ Δ. (See Example 3.29 for more details). These examples are quite informative about the great variety of locally compact abelian torsion groups surrounding the concept of divisibility here. We know that every abelian group A is contained in a certain divisible one, called a divisible hull (see [54], Proposition A1.33 and Corollary A1.36). This applies in particular to any compact abelian torsion p-group which is of the form presented in Proposition 41. The question is whether the divisible hull then has the structure of a locally compact abelian divisible torsion p-group. This is the case as the following proposition shows, which is not restricted to torsion groups at all. Proposition 44. Let A be a locally compact abelian p-group and D an algebraic divisible hull containing A. We give D the unique group topology for which A is an open subgroup. Then D is a locally compact abelian p-group. (See Proposition 3.42.) Conversely, every locally compact abelian divisible torsion p-group D possesses compact open subgroups (all of which are commensurable), and if C is one of these, then C has a locally compact abelian torsion p-group as divisible hull C ∗ in D, so that D = C ∗ ⊕ D󸀠 for some discrete divisible abelian torsion p-group D󸀠 which is a direct sum of copies of ℤ(p∞ ). (See [54], Theorem A1.42.) This gives an impression of the role of locally compact abelian divisible torsion p-groups. We shall look at the torsion-free analog in greater detail below. While the structure theorem of compact abelian torsion groups Proposition 41 looks quite satisfactory at first sight, it does not answer all questions we need to answer for our purposes in this book. Indeed the proof of the result we need is comparatively nontrivial, since it needs a selection process for which we resort to the compact Chabauty space attached canonically to each locally compact group. Theorem 45. In a compact torsion p-group every finite subgroup is contained in a finite (algebraic and topological) direct summand of the same p-rank. (See Theorem 3.34.)

On the structure of locally compact abelian groups | XXXV

Let us now look at the torsion-free counterpart. The divisible hull of a torsion-free abelian group A may be identified with ℚ ⊗ A where we may consider A as a subgroup identified with the image of the injection a 󳨃→ 1 ⊗ a : A → ℚ ⊗ A. In the present context of locally compact abelian p-groups we note that compact torsion-free p-groups have just one cardinal as an isomorphy invariant. Indeed the character group of a compact ̂ and thus is ≅ ℤ(p∞ )(J) for some torsion-free p-group G is a discrete divisible p-group G J set J as we noted before, and so G ≅ ℤp (see [54], Corollary 8.5). For locally compact torsion-free p-groups we shall prove the following theorems. Theorem 46. A locally compact abelian torsion-free p-group G is an open subgroup of its divisible hull DG = ℚ⊗G which is a torsion-free divisible p-group of p-rank ℵ for some cardinal ℵ. Moreover, DG = ℚ ⊗ C for any compact open subgroup C of G with C ≅ ℤℵ p and DG is a subgroup of ℚℵ p. (See Theorem 3.51.) Theorem 47. Every closed divisible subgroup of a torsion-free locally compact abelian p-group is a direct summand (algebraically and topologically). (See Theorem 3.62.) In the hierarchy of locally compact abelian torsion-free p-groups loc,ℕ ℤℕ ⊆ Dℤℕp = ℚ ⊗ ℤℕ p ⊆ (ℚp , ℤp ) p ,

all contained in the torsion-free group ℚℕ p that is not a p-group, provides us with a spectrum of examples. (See Examples 3.43 and 3.55.) What is remarkable here is that the maximal divisible subgroup D of (ℚp , ℤp )loc,ℕ is proper and dense; indeed it contains the dense divisible subgroup ℚ(ℕ) p . Indeed in Example 3.55 we analyze the maximal divisible subgroup D is (algebraically) the direct sum of ℚ(ℕ) and the maximal p

(ℕ) divisible subgroup of ℤℕ ≅ ℤℕ /ℤ(ℕ) , which we describe fairly explicitly. From p /ℤp all of this information let us keep in mind that in a locally compact abelian torsion-free group the maximal divisible subgroup need not be closed in spite of the two reasonably satisfactory Theorems 46 and 47. Let us now focus on the p-group ℚp itself and on its role as a (necessarily closed) subgroup of a locally compact abelian p-group. Perhaps such an inquiry is guided by the behavior of the locally compact abelian group ℝ as a subgroup of topological groups as it is reflected by the vector space splitting Theorem 36. By constructing examples presented in Corollary 3.22 and Theorem 3.84 we arrive at a surprising conclusion.

Theorem 48. There exists a second countable locally compact abelian p-group containing a closed subgroup isomorphic to ℚp which is not a direct summand algebraically and topologically.

XXXVI | Overview This means, dually, there are locally compact abelian p-groups with factor groups isomorphic to ℚp which do not split. In fact, we first construct such groups in Theorem 3.17 and in Example 3.81 and then conclude Corollary 3.22, respectively, Theorem 48 by way of duality. These results illustrate that the central role of ℝ, the archimedian completion of ℚ, in much of locally compact group theory, being illustrated by the vector space splitting Theorem 36, is starkly contrasted by the behavior of the p-adic completions ℚp of ℚ in the theory of periodic locally compact groups. Let us conclude this outlook on locally compact abelian p-groups by observing that things progress much more along expected lines under the assumption of finite rank. Theorem 49 (Čarin). A locally compact abelian p-group G is of finite p-rank iff it is of ∞ n the form G = ℚkp ⊕ ℤm p ⊕ ℤ(p ) ⊕ F for some nonnegative integers k, m, n and a finite p-group F, and the p-rank of G is precisely rankp G = k + m + n + rankp (F). (see Theorem 3.97.) This theorem reflects more closely a p-adic parallel to the vector space splitting Theorem 36.

Scalar automorphisms Among the methods we are using, the specification of scalar automorphisms of a periodic locally compact abelian group will be prominent. Remark 50. Every locally compact abelian p-group A is a natural ℤp -module, and in the case of a periodic locally compact abelian group loc

A = ∏ (Ap , Cp ) p∈π

it is a natural ̃ = ∏ ℤp ℤ p∈π

module by componentwise scalar multiplication z⋅g = (zp )p∈π ⋅(gp )p∈π = (zp ⋅gp )p∈π . ̃ is the profinite completion of the ring ℤ of integers. The compact ring ℤ Lemma 51 (The scalar morphism lemma). For a continuous automorphism α of a periodic locally compact abelian group G the following conditions are equivalent:

Scalar automorphisms | XXXVII

(1) (2) (3) (4)

α(H) ⊆ H for all closed subgroups H of G, α(⟨g⟩) ⊆ ⟨g⟩ for all g ∈ G, α(g) ∈ ⟨g⟩ for all g ∈ G, ̃ such that α(a) = r⋅a for all g ∈ G. there is an r ∈ ℤ

(See Lemma 4.7.) We note in passing that the first three conditions are equivalent in any locally compact group. Definition 52. An automorphism α ∈ Aut(A) of a periodic locally compact abelian group A satisfying the equivalent conditions of the preceding Lemma 51 is called a scalar automorphism. The group of all scalar automorphisms is written SAut(A). ̃ × , the group of invertible elements of ℤ, ̃ we denote the function a 󳨃→ r⋅a : For r ∈ ℤ A → A by μr ∈ SAut(A). Proposition 53. Assume that A is a locally compact abelian periodic group. Then ̃ × → SAut(A) is a quotient morphism of compact groups. In particular, (i) r 󳨃→ μr : ℤ SAut(A) is a profinite group and thus does not contain any nondegenerate divisible subgroups. (ii) The following conditions are equivalent: (a) SAut(A) = {idA , −idA }, (b) the exponent of A is 2, 3, or 4. In particular, A has exponent 2 if and only if −idA = idA . In the process of these discussions, we recover in our framework the following theorem of Mukhin [79]. Theorem 54. Let G be a locally compact abelian group written additively. Then ̃×, (a) SAut(G) is a homomorphic image of ℤ (b) if G is not periodic, then SAut(G) = {id, −id}, (c) if G is periodic, then SAut(G)= ∏p SAut(Gp ), where SAut(Gp ) may be identified with the group of units of the ring ℛ(Gp ) of scalars of Gp , namely, ℤp × ℤ(p − 1), { { { { { { ℤ(pm−1 ) × ℤ(p − 1), { { { × ℛ(Gp ) ≅ {ℤ2 × ℤ(2), { { { { ℤ(2m−2 ) × ℤ(2), { { { { {{0},

if p > 2 and the exponent of Gp is infinite,

if p > 2 and the exponent of Gp is pm ,

if p = 2 and the exponent of G2 is infinite,

if p = 2 and the exponent of G2 is 2m > 2,

if p = 2 and the exponent of G2 is 2,

̃ × such that (d) an automorphism α of G is in SAut(G) iff there is a unit z ∈ ℤ (∀g ∈ G) α(g) = z⋅g = ∏ zp ⋅gp p

for z = ∏ zp , g = ∏ gp . p

p

XXXVIII | Overview (See Theorem 4.25.) The significance of Mukhin’s theorem for the structure theory of near abelian groups is visible in the very Definition 5 via Theorem 7.43. Indeed if G is a near abelian locally compact group with a base A, then the inner automorphisms of G induce a faithful action of the Π-procyclic factor group G/CG (A) ≅ H/(H ∩ Z(G)) upon the base A. The structure of G is largely determined by the structure of SAut(A) and therefore ̃ × of units of ℤ. ̃ by the group ℤ

The group of units of the profinite completion of the ring of integers and its prime graph ̃ × is more complex than it appears at first. Its Sylow theory or primary The group ℤ decomposition is best understood in graph theoretical terms. The same graph theory turns out to be almost indispensable for dealing with the Sylow structure of near abelian groups in general. The graphs that we use are all subgraphs of a “universal” graph (which we also call the “mastergraph”) and which is used precisely to describe the ̃ × . We discuss it in the following. Sylow theory of ℤ A bipartite graph consists of two disjoint sets U and V and a binary relation ℰ ⊆ (U ∪ V)2 such that (u, v) ∈ ℰ implies u ∈ U and v ∈ V. The elements of U ∪ V are called vertices and the elements of ℰ are called edges. Any triple (U, V, ℰ ) of this type is called a bipartite graph. An edge labeled graph is a quadruple (U, V, ℰ , λ) such that (U, V, ℰ ) is a bipartite graph and λ is a function λ : ℰ → L for some set L of labels. Labels could be numbers, or symbols like 0. Definition 55. The bipartite edge labeled graph 𝒢 = (U, V, ℰ , λ),

ℰ ⊆ U × V,

will be called the prime mastergraph or mastergraph for short, where (i) U = π × {1} ⊆ π × {0, 1}, (ii) V = π × {0} ⊆ π × {0, 1}, (iii) ℰ = {((p, 1), (q, 0)) : p = q or p|(q − 1)}, (iv) λ: ℰ → ℕ ∪ {0}, 0, if p = q, λ(((p, 1), (q, 0))) = { νp (q − 1), if p < q. We shall call the vertices in U the upper and those in V the lower vertices. The edges ((p, 1), (p, 0)), p ∈ π, are said to be vertical, all others are called sloping. (See Figure 1.) We say that e = ((p, 1), (q, 0)) is the edge from p to q. (See Figure 2.) (See Definition 4.33.)

Geometric properties of the mastergraph | XXXIX

2

3

5

7

2

3

5

7

2

3

5

7

2

3

5

7

Figure 1: Vertical and sloping edges.

2

3

5

7

211

2

3

5

7

211

Figure 2: The five vertices in U connected to the lower vertex with label “211” in V .

2

3

5

7

11

13

2

3

5

7

11

13

Figure 3: The initial part of the mastergraph.

Geometric properties of the mastergraph The prime master-graph can be drawn and helps in forming a good intuition of the combinatorics involved (see Figure 3). Definition 56. Let p and q be any primes. Then 󸀠

󸀠

󸀠

ℰp = {e : e = ((p, 1), (p , 0)) ∈ ℰ such that p = p or p|(p − 1)},

the set of all edges emanating downwards from the vertex (p, 1) ∈ U will be called the cone peaking at p. Further the set 󸀠

󸀠

ℱq = {e : e = ((q , 1), (q, 0)) ∈ ℰ such that q |(q − 1)},

the set of edges ending below in the vertex (q, 0) ∈ V, is called the funnel pointing to q.

XL | Overview (See Definition 4.35.) Both the cones and the funnels provide a partition of the set of edges. It is instantly clear that each funnel is finite, and so the funnels are not as important as the cones. The structure of a cone is more interesting than that of a funnel, as the following proposition shows. Proposition 57. Let p be any prime. Accordingly, in the graph 𝒢 , the cone ℰp is peaking at the upper vertex (p, 1), and for each natural number n, it contains an edge e = ((p, 1), (q, 0)) labeled νp (q − 1) ≥ n. In particular, ℰp contains infinitely many edges. (See Proposition 4.36.) All applications of prime graphs which we use in the structure theory of near abelian groups are subgraphs of this mastergraph. Since for any periodic locally compact abelian group A we have a canonical sur̃ × → SAut(A), we need explicit information on the primary jective morphism μ: ℤ ̃ × . We are now going to describe this structure in structure—or p-Sylow structure—of ℤ additive notation in terms of the prime mastergraph 𝒢 = (U, V, ℰ ). Definition 58. For each edge e ∈ ℰ from p to q we set ℤ2 ⊕ ℤ(2), { { { 𝕊e = {ℤp , { { νp (q−1) ), {ℤ(p

if p = q = 2, if 2 < p = q,

if p < q.

(See Definition 4.37.) × ̃× = ∏ We noted in Remark 50 that ℤ p∈π ℤp . This will be exploited to show for the multiplicative subgroup ̃ × ) ≅ ∏ 𝕊e (ℤ p e∈ℰp

that p 󳨃→ ℰp : π → 𝒞 is a bijection from the set of primes to the set 𝒞 of cones such that 𝒞 = ⋃p∈π ℰp in the mastergraph. Taking into account these matters and the fact that every arithmetic progression of natural numbers must contain infinitely many primes (Dirichlet’s theorem), we can summarize as follows. ̃ × of units of the universal procyclic compactification ℤ ̃ of Theorem 59. (i) The group ℤ the ring of integers ℤ is the product ̃ × ≅ ∏ 𝕊e ℤ e∈ℰ

The structure of the invertible scalar multiplications of an abelian group

| XLI

extended over the set ℰ of all edges of the mastergraph, where 𝕊e is the profinite group given in (4.12) above. (ii) Its p-Sylow subgroup is the subproduct extended over the cone peaking in p, i. e., ℤ ⊕ ℤ(2) ⊕ ∏q>2 ℤ(2ν2 (q−1) ), if p = 2, ̃ × ) = ∏ 𝕊e ≅ { 2 (ℤ p ℤp ⊕ ∏q>p ℤ(pνp (q−1) ), otherwise. e∈ℰp (iii) For each p ∈ π fixed, ̃ × ) ≅ ℤp ⊕ Tp , (ℤ p

̃×) where Tp = tor(ℤ p

and where Tp contains a ℤp -submodule algebraically isomorphic to ℤ(ℕ) p whose closure is isomorphic to ∏n∈ℕ ℤ(pn ). (See Theorem 4.38.)

The structure of the invertible scalar multiplications of an abelian group and their prime graph Now let A be a periodic locally compact abelian group; the Sylow structure of SAut(A) ̃ × → SAut(A), preserving the Syis now easily discussed: The quotient morphism μ: ℤ low structures, and the structure of SAut(A) described so far in Theorem 54 allow a precise description of the Sylow structure of SAut(A). We associate with A the bipartite graph 𝒢 (A) = (U, V, ℰ (A)) with U and V as in the mastergraph and with ℰ (A) = {e ∈ E : e = ((p, 1), (q, 0)) such that SAut(Aq )p ≠ {idA }},

and for fixed p we let ℰp denote the set of edges in ℰ emanating from (p, 1) and for q let ℱq be the set of edges terminating in (q, 0). For any edge e in 𝒢 (A) from p to q the label is 0, if p = q, λ(e) = { νp (q − 1), if p|q − 1. We recall that for each q-primary component Aq , the ring of scalars SAut(Aq ) is either cyclic of order qr , the exponent of Aq , if it is finite, and it is ≅ ℤq otherwise. Thus its q-primary component is ℤ(qr−1 ), ≅{ ℤq ,

if the exponent of Aq is finite, otherwise.

This will allow us to prove the following theorem, complementing Theorem 54.

XLII | Overview Theorem 60 (The Sylow Structure of SAut(A)). Let A be a periodic locally compact abelian group and SAut(A) = ∏p∈π SAut(A)p the p-primary decomposition of the profinite group SAut(A) = ∏e∈ℰ(A) 𝕊e . Then (i) The p-primary decomposition of SAut(Aq ) is (additive notation assumed) ∏ SAut(Aq )pe = ∏ 𝕊e (A),

e∈ℱq

e∈ℱq

and this group is isomorphic, in case p = 2, to {0}, if A2 has exponent ≤ 2, { r−2 ℤ(2 ) ⊕ ℤ(2), if A2 has infinite exponent, and in case p > 2, to {ℤ(qr−1 ) ⊕ ⨁e∈ℱq ℤ(peλ(e) ), if Aq has finite exponent qr , { ℤ ⊕ ⨁e∈ℱq ℤ(pλ(e) if Aq has infinite exponent. e ), { q

(ii) The structure of the p-primary component SAut(A)p of SAut(A) (in additive notation) is ∏ (SAut(Aqe ))p = ∏ 𝕊e (A)

e∈ℰp

e∈ℰp

if p = 2 and A2 has exponent ≤ 2, ∏e∈ℰp ℤ(2λ(e) ), { { { { λ(e) r r−2 { { {ℤ(2 ) ⊕ ℤ(2) ⊕ ∏e∈ℰp ℤ(2 ), if p = 2 and A2 has fin.exp. 2 >2, { { ≅ {ℤ2 ⊕ ℤ(2) ⊕ ∏e∈ℰp ℤ(2λ(e) ), if p = 2 and A2 has inf. exponent, { { { λ(e) r−1 { ℤ(p ) ⊕ ∏e∈ℰp ℤ(pe ), if 2 < p and Ap has exponent pr , { { { { λ(e) if 2 < p and Ap has infinite index. {ℤp ⊕ ∏e∈ℰp ℤ(pe ), Here the edges of 𝒢 (a) serve as the index set for a product decomposition SAut(A) = ∏e∈E(A) 𝕊e which refines the product decomposition SAut(A) = ∏p∈π SAut(Ap ) that is induced by the p-primary (or Sylow) decomposition A =

∏loc p∈π (Ap , Cp ). (See Theorem 4.43.) This theorem illustrates the usefulness of the prime graph 𝒢 (A) which elucidates the fine structure of SAut(A). In many instances, the prime graph is equally helpful in the discussion of the Sylow structure of any periodic near abelian locally compact group G (see Section 10.1).

The prime graph of a near abelian group Staying with a periodic near abelian group G which is A-nontrivial for a base group A, we investigate the interaction of the different Sylow subgroups in terms of the prime

Application 1: The classification of topologically quasihamiltonian groups | XLIII

graph defined as a subgraph of the mastergraph 𝒢 = (U, V, ℰ ) of Definition 4.32 as follows. Definition 61. Assume that G is a periodic near abelian A-nontrivial locally compact group with a base group A and write G = CG (A)H. A subgraph 𝒢G = (UG , VG , ℰG )

of the mastergraph 𝒢 is called the prime graph of G provided the following conditions are satisfied: (i) an edge e = ((p, 1), (q, 0)) of the mastergraph is an edge in EG if and only if [Hp , Aq ] ≠ {1}; this edge is written epq and is called an edge leading from p to q, (ii) (p, 1) is an upper vertex in UG iff (G/CG (A))p ≠ {1}, (iii) (q, 0) is a lower vertex in VG iff Aq ≠ {1}. (See Definition 10.1.) We have a much sharper conclusion. Theorem 62 (Structure theorem IV on periodic near abelian groups). Let G be a periodic A-nontrivial near abelian group. Let epq be an edge in 𝒢G from p to q. Then we have the following conclusions: (1) If p ≠ q, that is, epq is sloping, then p|(q − 1), but above all (C1 ) for x ∈ Gp \ CG (Aq ) the function a 󳨃→ [x, a] : Aq → Aq is an automorphism of Aq ; in particular, [x, Aq ] = Aq , (C2 ) Aq ∩ Z(G) = {1}. (2) If p = q, that is, epq is vertical, then there is a unit s ∈ ℤ×p and a natural number m

m

m ∈ ℕ such that [x, a] = ap s for all a ∈ Ap , that is, [x, Ap ] = Aqp , and Ap ∩ CG (Ap ) has an exponent dividing pm .

(See Theorem 7.20.) The theorem gives an impression of the circumstances in which the intersection Aq ∩ Z(G)q can be nontrivial: The lower q-vertex has to be isolated in the prime graph in such a case.

Application 1: The classification of topologically quasihamiltonian groups The following definition is due to F. Kümmich [69]. Definition 63. A topological group G is called topologically quasihamiltonian if XY = YX holds for any pair of closed subgroups X and Y of G. This is equivalent to saying that XY is a closed subgroup whenever X and Y are subgroups of G.

XLIV | Overview With the framework we have provided with a theory of near abelian locally compact groups, we can classify completely the class of periodic topologically quasihamiltonian locally compact groups. The classification proceeds in two steps: In a first step we classify all locally compact topologically quasihamiltonian p-groups, and in a second step we classify all periodic locally compact topologically quasihamiltonian groups in one fell swoop. For Step 1 we need the following definition. Definition 64. The groups Mn defined by generators and relations for n = 2, 3, . . . according to n−1

n

Mn := ⟨a, b | b2 = 1, b2

= a2 , bab−1 = a−1 ⟩

are called generalized quaternion groups. (See Definition 8.5.) These groups also satisfy the relations a4 = 1

and [a, b] = a2

and are fully characterized by the following explicit construction: Mn ≅

ℤ(4) ⋊ ℤ(2n ) , Δ

where ℤ(2n ) acts on ℤ(4) by scalar multiplication with ±1 and where Δ is generated by (s, t), s = 2 + 4ℤ and t = 2n−1 + 2n ℤ. (Cf. [57], Definition 5.8.) We note that M2 is (isomorphic to) the usual group of quaternions Q8 = {±1, ±i, ±j, ±k} of eight elements. Here is Step 1. Theorem 65. A locally compact p-group G is topologically quasihamiltonian if and only if G is near abelian with a base group A and an inductively monothetic p-group G/A and at least one of the following statements holds: (a) G is abelian, (b) there is a p-procyclic scaling group H = ⟨b⟩ such that G = AH and there is a natural s number s ≥ 1, respectively, s ≥ 2, if p = 2, such that ab = a1+p for all a in A; the group G is A-nontrivial, (c) p = 2 and G ≅ A2 × Mn , where A2 is an exponent 2 locally compact abelian group and Mn is the generalized quaternion group of order 2n+1 ; in this case, A = A2 × ⟨a⟩ ≅ ℤ(2)(I1 ) × ℤ(2)I2 × ℤ(4), with a as in Definition 8.5 for suitable sets I1 and I2 ; the group G is A-trivial. (See Theorem 12.5.)

Application 2: The classification of topologically modular groups | XLV

Except for p = 2 it turns out that topologically quasihamiltonian p-groups in the

general locally compact domain are the same thing as near abelian p-groups. For the

exceptional compact 2-groups that are near abelian but fail to be topologically quasihamiltonian, see [57].

Next comes Step 2 (see Theorem 12.11).

Theorem 66. Let G be a locally compact periodic topologically quasihamiltonian group. Then, for each p ∈ π(G), the set of p-elements Gp is a topologically quasihamiltonian

p-group, and there is a compact open subgroup Up in Gp such that G = Gν(G) is (up to

isomorphism) the local product of topologically quasihamiltonian p-groups loc

G ≅ ∏ (Gp , Up ). p∈π(G)

Conversely, every group isomorphic to such a local product is a topologically quasihamiltonian group.

This theorem is proved with the aid of our Theorem 34.

For nonperiodic abelian groups we give an algorithmic description of topological-

ly quasihamiltonian locally compact groups in Theorem 12.17.

Theorem 66 can be visualized in terms of its prime graph, that all connected com-

ponents are either vertical edges and its end points or are isolated vertices. If we allow

ourselves the identification of the connected components of the prime graph with the subgroups they represent, we could reformulate Theorem 66 as follows.

Theorem 67. Let G be a locally compact periodic topologically quasihamiltonian group. Then each connected component of the prime graph of G represents a normal p-Sylow subgroup and G is a local direct product of these subgroups. (See Theorem 12.11.)

Application 2: The classification of topologically modular groups The closed subgroups of a topological group G form a lattice w.r.t. inclusion “⊆” as

partial order, and for closed subgroups X and Y of G one considers the operations X ∨ Y := ⟨X ∪ Y⟩ and X ∧ Y := X ∩ Y.

Definition 68. A topological group G is called topologically modular if the lattice of

closed subgroups is modular, that is, satisfies the law X ∨(Y ∧Z) = (X ∨Y)∧Z whenever X is a closed subgroup of Z.

XLVI | Overview This is equivalent to saying that the lattice of closed subgroups does not contain a sublattice isomorphic to the following form: ∙ ∙ ∙ ∙ ∙ (See [97], Theorem 2.1.2.) It is instructive to spend some time here on an example due to Mukhin which shows that topologically modular groups can be tricky. Example 69. Let p be any prime and I any infinite set (e. g., I = ℕ), set E := ℤ(p), define Gj := E 2 , Cj := {0} × E for all j ∈ I, and set loc

G := E (I) × E I ≅ ∏(Gj , Cj ), j∈I

where we took the discrete topology on the direct sum E (I) and the product topology I I 2 I on E I . We shall identify G with ∏loc j∈I (Gi , Ci ) and E × E with (E ) . The natural injection 2 I I I ι: G → (E ) = E × E is continuous but is not an embedding, since it is not open onto its image. Let D := {(x, x) : x ∈ E} ⊆ E 2 and let I

Δ = DI = {(xj , xj )j∈I : xj ∈ E} ⊆ (E 2 ) ≅ E I × E I denote the respective diagonals. Then Δ is a closed subgroup of (E 2 )I and so ι−1 (Δ) = D(I) is a closed subgroup of G. We shall denote it by Y. This is a noteworthy and perhaps slightly unexpected fact in view of the density of E (I) in E I . We verify as an exercise that the subgroup Y is not only closed, but even discrete, since ι(Y) meets trivially every open subgroup {0} × E K for a cofinite subset K ⊆ I. Now the product E I is the projective limit of its finite partial products E F as F ranges through the directed set ℱ of finite subsets F of I. Accordingly, G ≅ E (I) × lim E F ≅ lim (E (I) × E F ). F∈ℱ

F∈ℱ

Let D2 = {(xj )j∈I ∈ E I : (∃c ∈ E)(∀j ∈ I) xj = c}. Now we consider the following subgroups of G: X := E (I) × {0}, Z := E (I) × D2 ,

Application 2: The classification of topologically modular groups | XLVII

whence X ⊆ Z. Then X ∨ Y = E (I) × E I = G and so (X ∨ Y) ∩ Z = Z on the one side, while Y ∧ Z = Y and so X ∨ (Y ∧ Z) = X ∨ Y = G. Hence X ∨ (Y ∧ Z) ≠ (X ∨ Y) ∧ Z. Therefore G is a locally compact abelian nonmodular group. (See Example 13.2.) The example shows that the limit of a projective system of a locally compact topologically modular group with proper bonding maps need not be a topologically modular group and that a local product of a collection of finite abelian modular groups may fail likewise to be a topologically modular group. On our way to Mukhin’s classification of locally compact topologically modular groups, let us first describe the structure of an abelian topologically modular p-group. With the understanding of finite p-rank to be defined in Definition 3.89 we report Mukhin’s theorem on the classification of topologically modular locally compact abelian groups. Theorem 70 (Mukhin; see [78]). A locally compact abelian p-group G is topologically modular if and only if it satisfies one of the following conditions: (a) the torsion subgroup T of G is a discrete subgroup and rankp (G/T) is finite, (b) G contains a closed subgroup D such that G/D is compact and rankp (D) is finite. (See Theorem 13.11.) ̂ satisfies Notice that in the case of a locally compact abelian p-group G, the dual G ̂ is topologically (b) if G satisfies (a) and vice versa. Thus G is topologically modular iff G modular. A first step in the classification of nonabelian topologically modular groups is the case of p-groups. Proposition 71. Let G be a compactly ruled p-group. Then the following statements are equivalent: (1) G is a topologically modular group, (2) G is a strongly topologically quasihamiltonian group. (See Proposition 14.24.) We return to discussing nonabelian topologically modular groups. In contrast, however, with topologically quasihamiltonian groups, the normal p-Sylow subgroups are not the only building blocks of topologically modular groups. There is one additional category of building blocks which in the discrete situation have been known since the pioneering work of Iwasawa in the forties of the last century; see, e. g., [61, 62, 63].

XLVIII | Overview

Iwasawa (p, q)-factors Example 72. For a prime q let A be an additively written locally compact abelian group of exponent q and be either compact or discrete. Thus, algebraically, A is a vector space over the field GF(q). Now let p be a prime such that p|(q − 1). Then the multiplicative group of GF(q) contains a cyclic subgroup Z of order p. Let C = ⟨t⟩ be any p-procyclic group (that is, C ≅ ℤ(pk ) for some k ∈ ℕ or C ≅ ℤp ) and let ψ: C → Z be an epimorphism. Then C acts on A via r∗a = ψ(r)⋅a. Since Z is of order p, the kernel of ψ is an open subgroup of C of index p. Set G = A ⋊ψ C, the semidirect product for the action of C on A. Then A := A × {1} is a base subgroup of the near abelian locally compact group G, and H = ⟨(0, t)⟩ = {0}×C is a procyclic scaling p-subgroup. There are many maximal p-subgroups of G, namely, each ⟨(a, t)⟩ for any a ∈ A, and there is one unique maximal q-subgroup which is normal, namely, A. The simplest case arises when we take for C the unique multiplicative cyclic subgroup Sp (C) of order p itself, in which case we have G ≅ A⋊ℤ(p) and the set of elements of order p is A × (ℤ(p) \ {0}) and the set of q-elements is A × {0}. (See Example 13.15.) The class of locally compact near abelian topologically quasihamiltonian groups described in Example 72 is relevant enough in our classification to deserve a name. Definition 73. A locally compact group G which is isomorphic to a semidirect product A ⋊ψ C as described in Example 72 will be called an Iwasawa (p, q)-factor. The primes p and q are called the primes of the factor G. (See Definition 13.16.) The prime graph 𝒢 of an Iwasawa (p, q)-factor is one sloping edge epq with its endpoints. We would like to see an abstract characterization of a (p, q)-factor. For the purpose of presenting one, let us formulate some terminology for an automorphic action (h, a) 󳨃→ h⋅a : H ×A → A inducing a morphism α: H → Aut(A), α(h)(a) = h⋅a. If H/ ker α is an abelian group of order p for a prime number q, we shall say that the action of H on A is of order p. If H is a subgroup of a group G and A is a normal subgroup of G, then H acts on A via h⋅a = hah−1 . If this action is of order p, we say that H induces an action of order p on A. Proposition 74. Let p and q be primes satisfying p|(q − 1). A near abelian group G is an Iwasawa (p, q)-factor if and only if it satisfies the following conditions: (a) A = G󸀠 is an abelian group of exponent q; it is an either compact or discrete subgroup of G;

Iwasawa (p, q)-factors | XLIX

(b) there is a scaling group H which is a procyclic p-group; it induces an action of order p on A. If these conditions are satisfied, then G = A ⋊ H is a semidirect product and Z(G) = {hp : h ∈ H}. (See Proposition 13.17.) The significance of the (p, q)-factors for our classification is due to the following fact, which requires a technical proof that is not exactly short. Proposition 75. Let G = AH be an Iwasawa (p, q)-factor and A a topologically modular abelian group. Then G is a topologically modular group. (See Proposition 13.18.) Since a nondegenerate (p, q)-factor does not meet the criteria of a locally compact topologically quasihamiltonian group in Theorem 66, this allows us to remark a significant difference between topologically quasihamiltonian and topologically modular groups. Corollary 76. Any nondegenerate Iwasawa (p, q)-factor provides a topologically modular group which is not topologically quasihamiltonian. (See Corollary 13.19.) After a thorough discussion of compactly ruled topologically modular groups, using much of the information accumulated on near abelian groups we arrive at the following classification of compactly ruled locally compact topologically modular groups. Theorem 77 (Main theorem: topologically modular groups). Assume that G is a compactly ruled topologically modular group. Then π is a disjoint union of a set J of (i) sets σ of prime numbers which are either empty, or (ii) singleton sets σ = {p} such that Gσ is a normal p-Sylow subgroup and an Iwasawa p-factor, or (iii) two element sets σ = {p, q} such that for p < q the set Gσ is a normal σ-Sylow subgroup and an Iwasawa (p, q)-factor, such that loc

G = ∏(Gσ , Cσ ) σ∈J

for a family of compact open subgroups Cσ ⊆ Gσ . In particular, G is a periodic near abelian locally compact group. Conversely, every near abelian locally compact G of this form is a locally compact topologically modular group. (See Theorem 13.31.) We notice that, in the prime graph of G, the Sylow p-subgroups Gp constitute the connected components of either isolated vertices or vertical edges with its endpoints,

L | Overview while the Sylow subgroups G{p,q} which are Iwasawa (p, q)-components are connected components consisting of sloping edges with their endpoints. Moreover, every prime graph having such connected components can be realized as the prime graph of a periodic locally compact topologically modular group.

Application 3: The classification of strongly quasihamiltonian groups For a topological group G we denote by 𝒮𝒰ℬ(G) the set of all closed subgroups. In this last portion of the book we complete the classification of locally compact groups, in which for any closed subgroups X and Y of G the subset XY := {xy : x ∈ X, y ∈ Y} is a closed subgroup, that is, (∀X, Y ∈ 𝒮𝒰ℬ(G)) XY ∈ 𝒮𝒰ℬ(G).

(1)

In doing so we answer a question raised in 1984 by Yu. Mukhin in the Kourovka Notebook (cf. [66, Problem 9.32]). Call a locally compact group G strongly topologically quasihamiltonian if the above condition (1) is satisfied. Locally compact abelian groups with this property have been classified by Yu. Mukhin in [78]. In [66, Problem 9.32] he formulates the problem of classifying all strongly topologically quasihamiltonian groups. It follows immediately from the definitions that every strongly topologically quasihamiltonian group is also topologically modular. However, any finite Iwasawa (p, q)-factor illustrates the fact that not every topologically modular group is strongly topologically quasihamiltonian. (See Example 72.) R. Dedekind classified all finite groups G in which every subgroup is normal; see [23]. Such groups are usually termed hamiltonian; moreover, it can be easily shown that condition (1) holds in every hamiltonian group G. It was K. Iwasawa who termed a group G satisfying condition (1) quasihamiltonian and provided the classification of all (locally) finite quasihamiltonian groups in [61, 62]. Independently, G. Zappa and Gh. Pic obtained results in [115, 116, 89]. Corrections of original proofs came from F. Napolitani, see [84], and M. Suzuki, see [103]. Mukhin observed in [79] that not every topologically quasihamiltonian group is a strongly topologically quasihamiltonian group, and his example will be reproduced below in Lemma 79. Already Kümmich had observed in [68] that any topologically quasihamiltonian group is abelian, provided it has nontrivial connected components. What makes the determination of the class of strongly topologically quasihamiltonian groups difficult is the fact that it is not closed under forming projective limits; see Remark 80. Nevertheless, the class does have at least some good preservation properties which we record in the following.

Application 3: The classification of strongly quasihamiltonian groups | LI

Proposition 78. The class of strongly topologically quasihamiltonian groups is closed under (a) passing to closed subgroups and (b) passing to factor groups modulo closed normal subgroups. (See Proposition 14.2.) Let us first turn to the abelian situation. The following fact has already been observed in [78]; see also Example 69. As long as we are dealing with abelian groups in this section we use additive notation. Lemma 79. Let (pi )i∈I be a family of prime numbers and set G := ⨁ ℤ(pi ) × ∏ ℤ(pi ). i∈I

i∈I

Then G is strongly topologically quasihamiltonian if and only if I is finite. Remark 80. Several remarks are in order. (a) Letting, in the preceding Lemma 79, I be an infinite set yields an example of a topologically quasihamiltonian group which is not strongly topologically quasihamiltonian. (b) If I is infinite and the primes pi are pairwise different, then G is topologically quasihamiltonian and possesses the discrete subgroups X and Y with X +Y a nonclosed subgroup. Note that G is the projective limit of discrete quasihamiltonian groups. (c) The example in (b) is topologically modular (cf. Theorem 77) and, being abelian, is topologically quasihamiltonian, but it is not strongly topologically quasihamiltonian. For a classification theorem of abelian periodic strongly topologically quasihamiltonian groups, recall Lemma 3.91 explaining how the finite p-rank of a locally compact abelian p-group is exploited. Theorem 81. A necessary and sufficient condition for a locally compact abelian torsion group A to be strongly topologically quasihamiltonian is that there is a partition π(A) = κ ∪ ϕ and all of the following holds: (i) the set of primes ϕ is finite and Aϕ = Dϕ ⊕ Vϕ for an inductively monothetic summand Dϕ and a compact summand Vϕ ; for every p ∈ ϕ the p-primary component Dp of Dϕ has finite p-rank, (ii) Aκ is a discrete subgroup of A.

LII | Overview (See Theorem 14.21.) With the help of the preceding theorem we can complete Mukhin’s classification of abelian strongly topologically quasihamiltonian groups. Theorem 82. For a locally compact abelian periodic group A and open compact subgroup U the following statements are equivalent: (A) A is strongly topologically quasihamiltonian, (B) there is a partition of π(A) into disjoint subsets κ, ϕ, and μ := π(A) \ (κ ∪ ϕ) and all of the following holds: (i) κ = {p ∈ π(A) : Ap ∩ U = {0}} and Aκ is a discrete subgroup of A, (ii) ϕ = {p ∈ π(A) \ κ : rankp (Ap ) ≥ 2}; the set ϕ is finite and for all p ∈ ϕ the p-Sylow subgroup Ap is strongly topologically quasihamiltonian, (iii) Aμ is inductively monothetic, (iv) A = Aκ × Aϕ × Aμ topologically and algebraically. (See Theorem 14.22.) We are now ready for classifying strongly topologically quasihamiltonian groups that are split extensions of a base and a scaling subgroup (see Proposition 19). Theorem 83. Let the topologically quasihamiltonian group G be a semidirect product G = A ⋊ H of a strongly topologically quasihamiltonian base A and Π-procyclic scaling subgroup H. Then G is strongly topologically quasihamiltonian if and only if π(G) admits a partition π(G) = ϕ ∪ γ ∪ σ ∪ δ and the following conditions hold: (a) ϕ is finite, (b) Hγ is compact, (c) Gσ = Hσ , i. e., Aσ = {1}, (d) Gδ is discrete, (e) G = Gϕ × Gγ × Gσ × Gδ algebraically and topologically. (See Theorem 14.28.) Finally, in the nonperiodic case the following reduction can be stated. Theorem 84. Let G be a locally compact nonperiodic group. Then the following statements are equivalent: (A) G is strongly topologically quasihamiltonian, (B) G is near abelian with a base subgroup A that is a strongly topologically quasihamiltonian torsion group. (See Theorem 14.39.)

Application 3: The classification of strongly quasihamiltonian groups | LIII

Theorem 85. A nonperiodic locally compact group is strongly topologically quasihamiltonian, if and only if it is both topologically quasihamiltonian and topologically modular. (See Theorem 14.40.)

|

Part I: Background information on locally compact groups

Introduction Chapter I provides a background on totally disconnected locally compact groups. For example, results on the Chabauty topology are reviewed (see, e. g., Proposition 1.22) and Section 1.3 discusses semidirect product decompositions. Chapter 2 generalizes the Sylow theory known for profinite groups to periodic locally compact groups. Of independent interest might be a generalization of the Schur–Zassenhaus theorem; see Theorem 2.21 on compactly ruled groups. As a consequence we derive a fixed point theorem, Theorem 2.40, and obtain with its help Proposition 2.42, a quite general version of Maschke’s theorem. After this, Theorem 2.53 shows that in any compactly ruled group G a certain generalization of the nilpotent radical is a direct factor of G. Chapter 3 is devoted to studying periodic abelian groups—locally compact abelian groups consisting of compact elements only and being zero-dimensional—and we depart from Braconnier’s theorem (see Theorem 3.3), which says that such a group is a local product of its primary subgroups. Other results, not easily accessible in books on locally compact abelian groups, like splitting divisible subgroups as a direct summand, are treated. In Chapter 4, as a preparation for dealing with near abelian groups, we study automorphisms that leave invariant every closed subgroup of a given periodic locally compact abelian group—we term them scalar automorphisms and, for a locally compact abelian group A, the group of scalar automorphisms will be denoted by SAut(A). A graph theoretical device, the mastergraph, will display the various arithmetic features of scalar automorphisms. Then we shall re-prove a result of Yu. Mukhin from 1975 in Theorem 4.25 about describing the structure of SAut(A). Chapter 5 introduces and discusses in detail locally compact abelian groups for which any finite subset generates a monothetic subgroup. We term those groups inductively monothetic. Of interest are those inductively monothetic groups with compact primary subgroups— we call them Π-procyclic. The question if an extension of a locally compact abelian group A by a Π-procyclic group admits a Π-procyclic supplement is answered affirmatively for compact A in Theorem 5.41 and for open A in Theorem 5.32. During its proof extensive use of the compactness of the Chabauty topology will be made.

Generalities on the notation Unless we expressly state the contrary, we adhere to the following conventions. All groups in this book are locally compact and all homomorphisms continuous. We use the notation from [54]. A profinite, respectively, procyclic group is the projective limit of finite, respectively, finite cyclic groups. Thus a compact group is profinite if and only if it is totally disconnected and procyclic if and only if it is monothetic and totally disconnected. In Chapters 2 and 3 we shall call a topological group prosolvable if it is the projective limit of finite solvable groups; this is consistent with [93] but not with

4 | Introduction [52], where a projective limit of solvable topological groups is called prosolvable. For any topological group G, by G󸀠 we shall denote the closure of the subgroup generated by the set of all commutators. For a subset S of a set X we often shall denote by S󸀠 = X\S the set theoretic complement. For p a prime the field with p elements will be denoted by GF(p).

1 Locally compact spaces and groups 1.1 The presence of many compact open subgroups In a topological group G, an element g ∈ G is called a compact element if the monothetic subgroup ⟨g⟩ it generates is compact. We shall denote the set of all compact elements in G by comp(G) (cf., e. g., [54], Definition 7.44). Let G be a locally compact group. Then Weil’s lemma says for any element g ∈ G that either g ∈ comp(G) or else ⟨g⟩ is isomorphic to ℤ and therefore is, in particular, discrete (see, e. g., [54], Proposition 7.43). In a locally compact group G, the set comp(G) is known to be open-closed if the identity component G0 of G is compact, hence notably if G is totally disconnected. (See, e. g., [58].) Remark 1.1. We note in passing that G0 is compact if and only if G contains no isomorphic copy of ℝ; see, e. g., [52], Theorem 12.81. The openness of comp(G) is a consequence of the following known lemma for which we add a simple proof. Lemma 1.2. If G0 is compact, then every compact subgroup is contained in a compact open subgroup. Proof. Since G0 is compact, there exists a compact open subgroup U. Now let K be a compact subgroup. There is a finite symmetric subset R of K such that UK = UR. Induction shows that URk+1 = URk R = UKR = UK = UR. Then T := ⋂k∈K U k = ⋂x∈UK U x = ⋂r∈R U r is an open compact K-invariant subgroup of G and therefore TK is an open subgroup of G containing K. In the context of the present text, we shall introduce a remarkably powerful concept which is based on the following equivalence of basic properties. Proposition 1.3. For a topological group G the following statements are equivalent: (1) G is a directed union of compact open subgroups, (2) G is locally compact and every compact subset of G is contained in a compact subgroup, (3) G is locally compact and every finite subset of G is contained in a compact subgroup. Proof. For the proof of the equivalence of (2) and (3) we refer to Lemma 2.3 of the paper [13] of Caprace. So we shall argue the equivalence of (1) and (2). Statement (1) implies (2): Let K ⊆ G be compact. By (1) there is a directed set 𝒰 of compact open subgroups covering K, and since K is compact, a finite set U1 , . . . , Un ∈ 𝒰 suffices. Since 𝒰 is directed, there is a U ∈ 𝒰 containing all of U1 , . . . , Un and therefore containing K. https://doi.org/10.1515/9783110599190-001

6 | 1 Locally compact spaces and groups Statement (2) implies (1): By (2), G cannot contain any subgroup isomorphic to ℝ and so by Remark 1.1 the identity component G0 is compact. Thus by Lemma 1.2 every compact subgroup is contained in a compact open one. Let 𝒰 be the set of all compact open subgroups. If U, V ∈ 𝒰 , then U ∪ V is compact, and thus by (2) there is a W ∈ 𝒰 containing U ∪ V. Hence 𝒰 is directed. Now let g ∈ G. Then K := {g} is compact, so it is contained in a compact open subgroup, that is, there is a U ∈ 𝒰 such that g ∈ U. Hence G = ⋃U∈𝒰 U. This proves (1). Definition 1.4. A locally compact group G is said to be compactly ruled or, equivalently, topologically locally finite if the equivalent conditions of Proposition 1.3 hold. If G is compactly ruled with respect to the discrete topology, then G is locally finite; cf. [64]. In a compactly ruled group we have clearly G = comp(G). One observation is in order right away. The converse implication fails badly as any finitely generated infinite discrete torsion group shows. Such groups were constructed in 1964 by E. S. Golod, and more accessible ones later by R. I. Grigorchuk and, independently, as well by N. D. Gupta and S. Sidki (cf., e. g., [4]). Even more striking examples are the so-called Tarski monsters, constructed by A. Yu. Ol’shanskii in 1981. Remark 1.5. For almost every prime p, i. e., for p > 1075 , there exists a finitely generated infinite simple group G such that every proper subgroup of G is cyclic of order p. (See [85].) On the other hand, for relatively large classes of locally compact groups the relation G = comp(G) implies that G is compactly ruled. We provide some information now. Lemma 1.6. For a locally compact group G consider the following conditions: (1) G is compactly ruled, (2) G = comp(G) and the set 𝒰 of compact open subgroups is directed, (3) G = comp(G). Then (1)⇔(2)⇒(3), and if G is abelian, all three conditions are equivalent. Proof. On the basis of the definitions we have immediately (1)⇔(2)⇒(3). So we assume (3). Then G0 is compact, and hence every compact subgroup is contained in a compact open one by Lemma 1.2, that is, ⋃U∈𝒰 U = G. In addition we now assume that G is abelian. Then the product of two compact open subgroups is compact open. Thus 𝒰 is directed and so (3) implies (2). It seems to be known that the equivalence (1)⇔(3) applies to wider classes than locally compact abelian groups. But in the absence of a suitable reference we had better provide additional information. First we recall a standard notation.

1.1 The presence of many compact open subgroups | 7

Definition 1.7. Let G be a locally compact group and N a normal closed subgroup, and consider the quotient group Q := G/N. We then say that G is an extension of N by Q. Lemma 1.8. Let G be any locally compact group and N a closed normal subgroup of G. Then, for every compact subset E of G/N, there is a compact subset Ẽ of G such that ̃ E = EN/N. Proof. Since G is locally compact there is an open identity neighborhood U such that U is compact. Since gUN/N is open in G/N for all g in G and since E is compact, there is a finite subset F of G such that FUN/N = ⋃f ∈F fUN/N ⊃ E. Therefore the compact set FUN/N contains E. Let ϕ: FU → G/N denote the restriction of the quotient map. ̃ = EN/N. ̃ Then Ẽ := ϕ−1 (E) ∩ FU is a compact subset of G such that E = ϕ(E) Proposition 1.9. The class of compactly ruled groups is closed under the following operations: (i) passing to subgroups, (ii) passing to quotients, (iii) creating extensions, and (iv) the formation of products (as long as they remain locally compact). Proof. (i): This follows readily from the definition that any subgroup of a compactly ruled group is compactly ruled. (ii): For a compactly ruled group G let f : G → Q := G/N be a quotient map and E a finite subset of Q. Then, there is a finite subset F of G such that f (F) = E. Since G is compactly ruled, the subgroup ⟨F⟩ is compact and so is f (⟨F⟩). This set contains E, and hence Q is compactly ruled. (iii): Let N be normal in G and f : G → Q := G/N again the quotient map. Assume that both N and Q are compactly ruled. Let E be a finite subset of G; we need to show that ⟨E⟩ is compact. Since Q is compactly ruled and f (E) is finite, some compact subgroup C of Q contains f (E), and so ⟨E⟩ ⊆ f −1 (C). It is therefore no loss of generality to assume that Q is compact. Then by Lemma 1.8 there is a compact subset K such that f (K) = Q. After replacing E by (E ∪K)(E ∪K)−1 and renaming the items, we assume that E is symmetric, contains 1, and satisfies f (E) = Q, that is, EN = G. Now for any e, f ∈ E there is e󸀠 ∈ E and n ∈ N such that ef = e󸀠 n. Hence n ∈ E 3 ∩N and therefore we see that E 2 ⊆ E(E 3 ∩N). Induction yields E k+1 ⊆ E(E 3 ∩ N)k for all k ∈ ℕ. Hence k

⟨E⟩ = ⋃ E k ⊆ E⋅ ⋃ (E 3 ∩ N) ⊆ E⟨(E 3 ∩ N)⟩. 1j0

j>j0

= lim σmj (d)⋅ lim σmj (e) = σ(d)σ(e). j≥j0

j≥j0

Thus σ: D → G is a morphism and satisfies the conditions we need to complete the proof. Theorem 1.34. Let G be a topological group with a compact open normal subgroup C. Then any algebraic homomorphism of f : D → G/C from a subgroup D ⊆ ℚ lifts to an algebraic homomorphism F: D → G such that f = q ∘ F for the quotient morphism q: G → C/C. We have D ↑ ↑ id ↑ ↑ ↓ D

F

󳨀󳨀󳨀󳨀→ 󳨀󳨀󳨀󳨀→ f

G ↑ ↑ ↑ ↑q ↓ G/C.

18 | 1 Locally compact spaces and groups Proof. The case that D is cyclic is trivial. Therefore we assume D to be noncyclic. It is no loss of generality to consider in place of G the subgroup q−1 (f (D)). So assume that f is surjective. On D we take the discrete topology. We consider the closed subgroup G∗ of D×G defined as {(d, g) : q(g) = f (d)} and we let C ∗ = {0}×C ≅ C denote its compact open subgroup isomorphic to C. We may and will identify the elements of G∗ /C ∗ with the elements (d, gC) ∈ D×(G/C) satisfying gC = f (d). Then f ∗ : D → G∗ /C ∗ , f ∗ (d) = (d, f (d)) is an isomorphism of discrete torsion-free groups. Now Theorem 1.33 applies (with G∗ in place of G and C ∗ in place of A) and shows the existence of a morphism F ∗ : D → G∗ such that f ∗ = q∗ ∘ F ∗ with the quotient morphism q∗ : G∗ → G∗ /C ∗ . We have quotient morphisms r: G∗ → G, r(d, g) = g and s: G∗ /C ∗ → G/C, s(d, f (d)) = f (d) such that there is a commutative diagram D ↑ ↑ id ↑ ↑ ↓ D

F∗

󳨀󳨀󳨀󳨀→ 󳨀󳨀󳨀󳨀 → ∗ f

G∗ ↑ ↑ q∗ ↑ ↑ ↓ G∗ C∗

r

󳨀󳨀󳨀󳨀→ 󳨀󳨀󳨀󳨀→ s

G ↑ ↑ ↑ ↑q ↓ G . C

Since sf = f , the morphism F: D → G defined as the composition r ∘ F ∗ satisfies the requirement of the theorem. ∗

2 Periodic locally compact groups and their Sylow theory The structure of procyclic, equivalently, compact totally disconnected monothetic groups is rather special. By Proposition 1.15, periodic locally compact groups are characterized by the fact that every element g generates a procyclic group ⟨g⟩. On the other hand, it is the rather explicit number theoretical structure of ⟨g⟩ that makes a generalization of the classical Sylow subgroup theory to periodic groups possible. Therefore we briefly review the structure of a procyclic group P (see, e. g., [54, Example 8.75]). The relation P = ⟨g⟩ is equivalent to the presence of a morphism f : ℤ → P with dense image and f (1) = g where P is compact totally disconnected. Dually, this ̂ into ℤ ̂ ≅ 𝕋 = ℝ/ℤ whose torsion submeans that ̂f maps the discrete torsion group P ∞ group is ℚ/ℤ = ⨁p ℤ(p ), summed over all primes. Thus the primary decomposition ̂ is ⨁ (P) ̂ p , where (P) ̂ p is a subgroup of ℤ(p∞ ), and thus is ℤ(pn ), n = 0, 1, 2, . . . , ∞. of P p Hence we have the following. Remark 2.1. Every procyclic group P is isomorphic to a product ∏p Pp of procyclic p-groups {1}, { { { Pp = {ℤ(pn ), { { {ℤ p ,

or for some natural number n, or the p-adic integers.

This remark permits the basic definitions that make Sylow subgroup theory possible for periodic locally compact groups. Definition 2.2. Let G be a locally compact group. (i) If g ∈ G such that ⟨g⟩ is procyclic then π(g) will denote the set of all primes such that the profinite abelian group ⟨g⟩ has a nontrivial p-primary subgroup according to Remark 2.1. If ⟨g⟩ ≅ ℤ we set π(g) = 0. For a subset S ⊆ G we write π(S) := ⋃g∈S π(g). (ii) For any set σ of primes, we say that g is a σ-element, provided π(g) is a subset of σ. We let Gσ denote the set of all σ-elements and say that a locally compact group G is a σ-group if it is periodic and π(G) ⊆ σ. (iii) A maximal σ-subgroup of a locally compact group G will be termed a σ-Sylow subgroup of G. If σ = {p} we shall simply call it a p-Sylow subgroup, consistent with the tradition in the theory of finite and profinite groups. Note that, according to these conventions, the identity element of G is a σ-element for every set of primes σ and that every element is a π(G)-element. https://doi.org/10.1515/9783110599190-002

20 | 2 Periodic locally compact groups and their Sylow theory Remark 2.3. What we call a σ-Sylow subgroup of a locally compact group G in Definition 2.2 (iii) has also been called a σ-subgroup; see, e. g., [60, p. 283, Definition 5.4] and [93, p. 35]. We observe right away that Zorn’s lemma implies the following. Lemma 2.4. Every σ-element and indeed every σ-subgroup in a locally compact group is contained in a σ-Sylow subgroup. A sequence of additional remarks follows straightforwardly. Remark 2.5. Let G be a locally compact group and g ∈ G. (a) If G is finite then π(g) is the set of prime divisors of the order of g. For g = 1 this set is empty. Moreover, π(G) is the set of prime divisors of the order of G. (b) If the filter basis 𝒩 of open normal subgroups of G converges to 1, then G = lim G/N is a projective limit of discrete groups. Then for any g ∈ G we have ←󳨀󳨀𝒩 π(g) = {p : (∃N ∈ 𝒩 ) p ∈ π(gN)}. (c) Strictly speaking, π(g) should be written πG (g) in order to indicate the dependence on G, which is clear in most circumstances. However, if g ∈ H = H ⊆ G, then ⟨g⟩ ⊆ H, and thus πH (g) = πG (g). So π(g) is determined by any closed subgroup of G to which g belongs. This is helpful if G is periodic and g ∈ G; then π(g) is determined by any compact (and so profinite) subgroup U of G to which g belongs which reduces us to situation (b). In the six-element group G := ℤ(3) ⋊ ℤ(2) ≅ S3 , ℤ(2) = {0, 1} mod 2ℤ, we have G2 = {(0, 0)} ∪ (ℤ(3) × {1}), which is not a subgroup. It is noteworthy that in this entire subject this group and its powers are the source of many informative counterexamples. See, e. g., Example 2.23. However, in this context, the following results will turn out to be useful. Lemma 2.6 (Closure lemma). Let G be a totally disconnected locally compact group and σ a set of primes. Then Gσ is closed. Proof. Let g ∈ Gσ . We have to show π(g) ⊆ σ. Let (gj )j∈J be a net of σ-elements converging to g ∈ G. We claim that p ∈ π(g) implies p ∈ σ. Suppose that ⟨g⟩ ≅ ℤ, that is, g ∉ comp(G). Since comp(G) is closed (see, e. g., [58]), we would have gj ∉ comp(G) for all sufficiently large j and ⟨gj ⟩ ≅ ℤ for these j, which means that gj cannot be a σ-element. Thus ⟨g⟩ is compact and thus procyclic since G is totally disconnected. Since G is totally disconnected, ⟨g⟩ is contained in a compact open subgroup U by Lemma 1.2, and we may assume by passing to a cofinal index set that all gj are contained in U. Thus (gi )i∈I is a net in the compact group U. It is therefore no loss of

2 Periodic locally compact groups and their Sylow theory | 21

generality to assume that the group G itself is compact and therefore profinite since it is totally disconnected. By Remark 2.1, the procyclic group M = ⟨g⟩ is the direct product of its q-primary components as q ranges through π(g). We claim that all these q are contained in σ. By way of contradiction assume that there is a q ∈ π(g) \ σ. Now Mq = ⟨h⟩ for some q-element h ∈ M. We know that M = ⟨g⟩ = limj ⟨gj ⟩ in 𝒮𝒰ℬ(G) (see [58]). So there is a net of elements hi ∈ ⟨gj(i) ⟩, i ∈ I, for a subnet (gj (i))i∈I such that h = limi hi (see, e. g., [39]). The profinite group G has a compact open normal subgroup N not containing h. Then hN = hi N for all sufficiently large i in the finite group G/N. But hN is a q-element while hi N = gj(i) N is a σ-element. This contradiction proves the lemma. Let G = 𝕋 = ℝ/ℤ; then the set of p-elements for any prime p is p1∞ ℤ/ℤ = ℤ(p∞ ), which is dense and not closed. The hypothesis of zero-dimensionality in Lemma 2.6 is therefore crucial. Corollary 2.7. In a totally disconnected locally compact group G all σ-Sylow subgroups are closed. Proof. Let S be a subgroup contained in the set Gσ of all σ-elements. By the closure Lemma 2.6, Gσ is closed. Therefore S is contained in Gσ and thus is a σ-group. Now assume that S is a σ-Sylow subgroup. Then by what we just saw, S is a σ-group. Then by the maximality of S we have S = S. Lemma 2.8. Every periodic locally compact group is generated by (the union of) its p-Sylow subgroups. Proof. Let g ∈ G. Then C := ⟨g⟩ ≅ ∏p∈π(C) Cp . Thus C is generated by the union of the Cp . Each Cp is contained in some p-Sylow subgroup Sp by Lemma 2.4. The lemma follows. In some situations, sharper results hold; see, e. g., Corollary 7.21. Lemma 2.9. Let G be a locally compact group with a closed normal subgroup N such that both N and G/N are σ-groups. Then G is a σ-group. Proof. Let g ∈ G. Assume that the p-primary component (⟨g⟩)p is nontrivial; we have to show p ∈ σ. We may assume that g is a p-element and ⟨g⟩ is a p-group. Then its image in G/N is either a nontrivial p-group or trivial. In the former case p ∈ σ since G/N is a σ-group. In the latter case, ⟨g⟩ ⊆ N and thus p ∈ σ since N is a σ-group. Lemma 2.10. Let G be a periodic locally compact group and assume that for σ ⊆ π(G) all σ-Sylow subgroups are conjugate. Let N be any normal subgroup. Then for any σ-Sylow subgroup S of G the intersection N ∩ S is a σ-Sylow subgroup of N. Proof. Let Nσ be a σ-Sylow subgroup of N. Then by Lemma 2.4 there is a σ-Sylow subgroup S∗ of G containing Nσ . By assumption, (S∗ )g = S for a suitable g ∈ G. Conju-

22 | 2 Periodic locally compact groups and their Sylow theory gation with g induces an automorphism on N and so Nσg is a σ-Sylow subgroup of N. Hence S ∩ N = (S∗ )g ∩ N = Nσg is a σ-Sylow subgroup of N as asserted. Lemma 2.11. Let f : G → H be a quotient morphism of locally compact periodic groups and σ a set of primes. Then f (Gσ ) = Hσ . Proof. The proof of f (Gσ ) ⊆ Hσ is straightforward. We show the reverse containment. Let h ∈ Hσ . Then ⟨h⟩ is a procyclic σ-group and the restriction of f to f −1 (⟨h⟩) is a quotient map onto ⟨h⟩. Thus we may assume that H = ∏p∈σ Hp with procyclic p-groups Hp . Let g be any element in f −1 (h). Then ⟨g⟩ is a compact monothetic group mapping onto H under f . Thus we may assume that G = ⟨g⟩ and that G = ∏p∈π(G) Gp with procyclic p-groups Gp . We may identify Gp in the obvious fashion with a subgroup of G; similarly with Hp . Then f (Gp ) ⊆ Hp for each p ∈ π(G), and thus f (Gp ) = {1} for p ∈ π(G) \ σ. The monothetic subgroup ∏p∈σ Gp finally is a σ-group mapping surjectively onto H, and thus every element of Hσ is in the image of G.

2.1 Normal σ-subgroups On the way to a study of normal Sylow subgroups let us record the observation that a normal σ-subgroup is contained in any σ-Sylow subgroup. Proposition 2.12. Let G be a periodic group and σ be a set of primes. Suppose that G contains a closed normal σ-subgroup N. Then the following statements hold: (i) If H is a closed σ-subgroup, then ⟨NH⟩ is a closed σ-subgroup of G. (ii) If H is any σ-Sylow subgroup, then N ⊆ H. (iii) If H is any σ-Sylow subgroup of G, then H/N is a σ-Sylow subgroup of G/N. Proof. (i) In light of Corollary 2.7 it suffices to show that NH does not contain nontrivial σ 󸀠 -elements. Thus, let x ∈ NH ∩ Gσ󸀠 . Then xN ∈ NH/N on the one hand is a σ-element by Lemma 2.11 since H ⊆ Gσ , and it is a σ 󸀠 -element on the other hand since x ∈ Gσ󸀠 . Hence xN = N, that is, x ∈ N. Thus x = 1 since N ⊆ Gσ and x ∈ N ∩ Gσ󸀠 . As we noticed in Remark 2.1, an arbitrary element g of NH is a product yx = xy of a σ-element y and a σ 󸀠 -element x, by the preceding observation g it is necessarily a σ-element. This proves claim (i). (ii) This assertion follows via (i) from the maximality of a Sylow subgroup. (iii) Let ϕ: G → G/N denote the quotient homomorphism. By Lemma 2.4 the σ-subgroup H/N is contained in a σ-Sylow subgroup S and so H ⊆ ϕ−1 (S). For proving equality we show that ϕ−1 (S) is a σ-group; then the fact that H is a σ-Sylow group and therefore a maximal σ-subgroup proves equality. So let x ∈ ϕ−1 (S). By Remark 2.1, ⟨x⟩ is a direct product of procyclic p-groups for different primes p, and it suffices to show that each of the primes p is contained in σ. So we assume that x is a p-element.

2.1 Normal σ-subgroups | 23

If ϕ(x) = 1, then x ∈ N ⊆ H ⊆ Gσ and we are done. Assume ϕ(x) ≠ 1. Now ϕ(x) is a p-element in S and so p ∈ σ proving the claim. The center Z(G) of a periodic locally compact group G, as an abelian periodic group, has a unique σ-Sylow subgroup Z(G)σ which is normal in G. By Lemma 2.4 it is contained in some σ-Sylow subgroup H of G, and by Proposition 2.12 (ii), it is indeed contained in every σ-Sylow subgroup H. The precise relation is described in the following proposition. Proposition 2.13. Let G be a periodic group and H be a σ-Sylow subgroup. Then H ∩ Z(G) = Z(G)σ . Proof. The intersection H ∩ Z(G), being a σ-subgroup of Z(G), is contained in Z(G)σ . The reverse conclusion follows by setting N := Z(G)σ in Proposition 2.12 (ii). The subsequent statements relate to Proposition 2.12. Definition 2.14. If a continuous action (γ, k) 󳨃→ k : Γ × K of topological groups is such that k 󳨃→ γ⋅k : K → K is an automorphism of K we say that the action is automorphic. Remark 2.15. If the topological group Γ acts automorphically on the topological group K, we can form the semidirect product G := K ⋊ Γ with the multiplication (k1 , γ1 )(k2 , γ2 ) = (k1 (γ1 ⋅k2 ), γ1 γ2 ) with the result that (1, γ)(k, 1)(1, γ)−1 = (γ⋅k, 1). Let Ig denote the inner automorphism x 󳨃→ gxg −1 of G. Then both K and Γ may be identified with closed subgroups of G in such a fashion that the action k 󳨃→ γ⋅k gets identified with Iγ |K , that is, the inner automorphism implemented by γ on G restricted to K. Lemma 2.16. Let H be a locally compact and compactly ruled group acting automorphically on a locally compact abelian periodic group N. Assume that for a set σ of primes, P is a closed σ-subgroup of H. Then Nσ ⋊ P ⊆ N ⋊ H is a closed σ-subgroup. Proof. We write N additively and note that Nσ is a fully characteristic subgroup of N (i. e., is invariant under all endomorphisms). Let x, y ∈ Nσ and h, k ∈ Hσ . Then (x, h)(y, k)−1 = (x − hk −1 ⋅y, hk −1 ). Now hk −1 ⋅y is a σ-element since y is a σ-element and Nσ is invariant under endomorphisms. Hence x − hk −1 ⋅y is a σ-element since Nσ is a subgroup of the abelian group N. As P is a closed subgroup of H by hypothesis, we know now that Nσ ⋊ P is a subgroup of G := N ⋊ H. Since Nσ is closed in the periodic group N by Lemma 2.6 and P is closed in H by hypothesis, the product Nσ ⋊ P is closed in N ⋊ H and is, therefore, a locally compact subgroup of G. For the proof that Nσ ⋊ P is a σ-group we now assume without loss of generality that N and H are σ-groups, and we must show that G is a σ-group. Since the group N is abelian and periodic, it is compactly ruled by Corollary 1.6. So G is compactly ruled by Proposition 1.9. Now Proposition 2.12 (i) applies to G = N ⋊ H and shows that G is a σ-group as asserted. The example of G := ℤ(3) ⋊ ℤ(2) shows that here prℤ(3) (G2 ) = ℤ(3) ≠ {1} = ℤ(3)2 .

24 | 2 Periodic locally compact groups and their Sylow theory Proposition 2.17. Let G be a locally compact periodic group with a closed abelian normal subgroup N and assume that H is a closed abelian subgroup such that G = NH. Consider a set of primes σ ⊆ π(G). Then the set Nσ Hσ is a closed σ-subgroup of G. Proof. We let D := N ∩ H. The function ν: ND ⋊ HD → GD , ν(aD, hD) = ahD is an isomorphism. By Section 1.3 (and indeed again by Proposition 1.31) we have two surjective morphisms μ: N ⋊ H → G, μ(a, h) = ah and δ: G → ND ⋊ HD ≅ GD , and by Proposition 1.31 (ii), the composition δ ∘ μ is the natural quotient morphism N ⋊H →

N H ⋊ . D D

(1)

Since G is periodic, the abelian locally compact subgroups N, H, and D are all periodic and thus have a Braconnier decomposition as in Theorem 3.3 (see Chapter 3). It follows that for σ 󸀠 = π(G) \ σ we have product decompositions N = Nσ × Nσ󸀠 ,

H = Hσ × Hσ󸀠 , D = Dσ × Dσ󸀠 ,

where Dσ = Nσ ∩ Hσ ,

Dσ󸀠 = Nσ󸀠 ∩ Hσ󸀠 . We deduce from these circumstances that

Nσ Nσ D Nσ ×Dσ󸀠 ≅ = and Dσ D Dσ ×Dσ󸀠 Hσ Hσ Dσ Hσ ×Dσ󸀠 ≅ = Dσ D Dσ ×Dσ󸀠

are locally compact groups; in particular, Nσ D Hσ D N H ⋊ is closed in ⋊ . D D D D

(2)

Since H is an abelian periodic locally compact group, it is compactly ruled by Corollary 1.6. So Lemma 2.16 applies and shows that the subset Nσ × Hσ of N ⋊ H is a subgroup and is contained in (N ⋊H)σ . Since μ is a morphism, it follows that the image Nσ Hσ = μ(Nσ ⋊ Hσ ) is a subgroup of G, and Lemma 2.11 shows that μ((N ⋊ H)σ ) = Gσ . Therefore the subgroup Nσ Hσ of G is contained in Gσ . It remains to prove the claim that the subgroup Nσ Hσ ⊆ G is closed. So we have before us the following commutative diagram of morphisms: N ⋊H ↑ ⊆↑ ↑ ↑ ↑ Nσ ⋊ Hσ

μ

󳨀󳨀󳨀󳨀→ 󳨀󳨀󳨀󳨀󸀠→ μ

G ↑ ⊆↑ ↑ ↑ ↑ Nσ Hσ

δ

󳨀󳨀󳨀󳨀→

N D

󳨀󳨀󳨀󳨀 → 󸀠

Nσ D D

δ

⋊ HD ↑ ↑ ⊆ ↑ ↑ ↑H D ⋊ Dσ .

(3)

2.2 Normal σ-Sylow subgroups | 25

All vertical maps are inclusion embeddings, and μ󸀠 = μ|(Nσ ⋊ Hσ ) and δ󸀠 = δ|(Nσ Hσ ). N D H D N D H D Since Dσ ⋊ Dσ is closed in ND ⋊ HD by (2), δ−1 ( Dσ ⋊ Dσ ) is a closed subgroup of G. The subset Gσ is closed in G by Lemma 2.6. We claim that (δ−1 (

N D H D Nσ D Hσ D ⋊ )) = δ−1 ( σ ⋊ σ ) ∩ Gσ = Nσ Hσ , D D D D σ

which will establish the claim and finish the proof. Proof of “⊇”: The function δ being the composition g󳨃→gD

ν−1

G 󳨀󳨀󳨀󳨀→ G/D 󳨀󳨀󳨀󳨀→ maps Nσ Hσ onto

Nσ H σ D D

ν−1

󳨀󳨀󳨀󳨀→

Nσ D D

⋊

Hσ D , D

N H ⋊ D D

and this proves “⊇”.

N D

H D

Proof of “⊆”: Let g ∈ Gσ be such that δ(g) = (aD, hD) ∈ Dσ ⋊ Dσ with a ∈ Nσ and h ∈ Hσ where ahD = ν(aD, gH) = gD ∈ G/D. Then there is a d ∈ D = N ∩H such that g = ahd. Since D = N ∩H we have D ⊆ Z(G), whence [ah, d] = 1. The σ-element g is contained in the profinite abelian group ⟨ah, d⟩ and both g and ah are σ-elements and thus d is a σ-element, that is, d ∈ Dσ ⊆ Hσ . Then hd ∈ Hσ and g = a(hd) ∈ Nσ Hσ . Hence “⊆” is proved.

2.2 Normal σ-Sylow subgroups Our earlier Proposition 2.12 indicates that one must investigate normal σ-Sylow subgroups N of a locally compact group G in a more systematic fashion. In order to have a meaningful Sylow theory for G, we shall consistently assume that G is periodic. Recall: If G is a locally compact group and σ a set of primes, then Gσ is the set of all σ-elements. (Cf. Definition 2.2.) Proposition 2.18. Let N be a maximal σ-subgroup of a locally compact periodic group G. Then the following conditions are equivalent: (1) N is a normal Sylow subgroup, (2) N = Gσ , (3) N is normal and σ ∩ π(G/N) = 0. Proof. Statement (1) implies (2): Since N is a σ-subgroup, the containment N ⊆ Gσ is trivial, and we have to prove the converse containment. So let g ∈ Gσ . We set M := ⟨g⟩. Let Mp be a p-primary component of the monothetic σ-group M for any p ∈ σ. By the normality of N and compactness of Mp , the product NMp is a closed subgroup. Since N is a maximal σ-subgroup and p ⊆ σ we have Mp ⊆ N. Since this holds for all p ∈ σ, we have g ∈ M ⊆ N and so g ∈ N which we had to show. Statement (2) implies (3): Since each endomorphism of the topological group G maps Gσ into itself, (2) shows that N is fully characteristic and so, in particular, normal. So we have a quotient morphism f : G → G/N. Suppose p ∈ σ ∩ π(G/N); we have

26 | 2 Periodic locally compact groups and their Sylow theory to derive a contradiction which will prove (3). Then there is a g ∈ G such that ⟨f (g)⟩ is a nontrivial procyclic p-group. The kernel of f is a σ-group, and so the full inverse image F of ⟨f (g)⟩ is a σ-group by Lemma 2.9. Hence F ⊆ Gσ . Since N ⊆ F, now (2) says N = F. Thus ⟨f (g)⟩ ≅ F/N is singleton, and this is the contradiction we were looking for. Statement (3) implies (1): By Lemma 2.4 there is a σ-Sylow group S with N ⊆ S and we have to show N = S. By way of contradiction we suppose that there is an element g ∈ S \ N. Recall that G is periodic. So ⟨g⟩ is procyclic. In view of the structure of ⟨g⟩ discussed in Remark 2.1 we may assume that g is a p-element for some p ∈ σ. Let f : G → G/N again denote the quotient morphism. Then ⟨f (g)⟩ is a nontrivial procyclic p-group. Therefore p ∈ π(G/N). However, this contradicts assumption (3). This contradiction shows N = S and thus (1). Lemma 2.19. Let G be a periodic group containing a closed σ-subgroup S and a closed σ 󸀠 -subgroup T such that G=ST. Then (1) S is a σ-Sylow subgroup and T is a σ 󸀠 -Sylow subgroup, (2) S ∩ T = {1}, (3) S = Gσ iff S is normal. Proof. (1) Let S∗ be a σ-Sylow subgroup containing S according to Lemma 2.4. Then, as S∗ ∩ T = {1}, making use of the modular law (for abstract groups), one finds S∗ = S∗ ∩ (ST) = S(S∗ ∩ T) = S. Likewise one observes that T is a σ 󸀠 -Sylow subgroup. (2) If g ∈ S ∩ T, then g is a σ-element and a σ 󸀠 -element as well. The only element with this property is 1. (3) This is immediate from Proposition 2.18.

2.3 A Schur–Zassenhaus theorem We pursue the idea of a normal σ-Sylow subgroup and all of its ramifications. Definition 2.20. Let G be a locally compact periodic group and N a closed subgroup. We say that N satisfies the Schur–Zassenhaus condition if and only if it satisfies the equivalent conditions of Proposition 2.18 for σ = π(N). The name of the Schur–Zassenhaus condition originates from a class of theorems whose basic hypothesis it is. The following theorem is a typical example. From Definition 1.32 we recall that complement is tantamount to what is sometimes also called a semidirect cofactor. See Section 1.3. Another terminology that is used in the literature for the existence of a semidirect cofactor for a normal subgroup is that the quotient

2.3 A Schur–Zassenhaus theorem

| 27

N → G → G/N splits. In various situations in the structure theory of locally compact groups where the so-called splitting theorems emerge the generic hypothesis in ever so many different ways states that N and G/N have “opposite” or at least “significantly different” properties. The Schur–Zassenhaus condition is one of these. Other examples are the case of a locally compact group G with a normal subgroup N isomorphic to ℝn and G/N being compact (see, e. g., [55], [54], Theorem 5.71)) or of a compact group G with the commutator subgroup N and the factor group by definition of N being abelian (see [54], Theorem 9.39), or of some splitting theorems in [51] that resemble the result we are about to present in the form of Theorem 2.21 or Theorem 2.31. Theorem 2.21. Let G be a periodic group and N a closed subgroup satisfying the following two conditions: (1) N satisfies the Schur–Zassenhaus condition, (2) G/N is a sigma-compact compactly ruled group. Then the following conclusions hold: (i) N possesses a complement H in G. (ii) Let K be a closed subgroup of G such that K ∩ N = {1} and assume that G/N is compact. Then there is a g ∈ G such that gKg −1 ⊆ H. Remark 2.22. The condition that G/N is compactly ruled is satisfied if G itself is compactly ruled. Proof. We need to prove the existence of a closed subgroup H such that G = NH and N ∩ H = {1}; then Proposition 1.27 will prove the claim; this is a typical application of that proposition. Moreover, for a closed subgroup K of G with N ∩ K = {1} we need to show that a conjugate of K is contained in H. We shall establish the existence of H and the conjugacy assertion through several steps, each one more general than the previous one: (a) G is compact, (b) G/N is compact, (c) G/N is sigma-compact. Step (a) is the content of [93, Theorem 2.3.15]. (b) Put Q := G/N. By Lemma 1.8 there exists a compact subset E of G such that EN/N = Q. Since G is compactly ruled the subgroup T := ⟨E⟩ is compact and hence Q ≅ TN/N ≅ T/T ∩ N holds by the open mapping theorem (see [54, Theorem EA1.21, pp. 704 f.]). From (a) the existence of a complement H for T ∩ N follows. Thus T = (T ∩ N)H and T/(T ∩ N) ≅ H. Therefore G = NH. Since H ≅ Q and π(Q) ∩ π(N) = 0 deduce that N ∩ H = 1. Now Proposition 1.27 implies that G = N ⋊ H. Assume that K is a closed subgroup satisfying N ∩ K = {1}. The projection C of K into N along H is compact since K ≅ NK/N ⊆ G/N is compact. Then C H = {h−1 ch : c ∈ C, h ∈ H} is a compact H-invariant subset of N. Since G is compactly ruled, M := ⟨C H ⟩

28 | 2 Periodic locally compact groups and their Sylow theory is a compact, H-invariant subgroup of N. Thus MH is a compact subgroup satisfying MH = M ⋊ H and containing K. Then (a) implies that a conjugate of K in MH is contained in H. This proves (b). (c) Since G/N is compactly ruled and sigma-compact, there is an ascending chain (Ui )∞ i=1 of open subgroups containing N whose union is G and which are such that Ui /N is compact for all i. Then (b) implies, for every i, the existence of a compact subgroup Hi of Ui such that Ui = N ⋊ Hi . Note for later that π(Hi ) = π(Ui /N) ⊆ π(Q).

(2.1)

Let us set H1∗ = H1 ⊆ Ui . Assume that Hi∗ ⊆ Ui has been constructed. Inductively we ∗ find elements gi ∈ Ui+1 such that gi Hi+1 gi−1 ⊇ Hi∗ in Ui+1 . Set Hi+1 := gi Hi+1 gi−1 in Ui+1 . Then H1∗ ⊆ H2∗ ⊆ ⋅ ⋅ ⋅

and (∀i ∈ ℕ) Ui = gi Ui gi−1 = gi NHi gi−1 = NHi∗ .

Put H := ⋃i Hi∗ . We claim that G = NH. Indeed, if x ∈ G, then there is i with x ∈ Ui = NHi∗ ⊆ NH; the latter group is contained in NH. Next note that H is closed since its intersection with each of the open subgroups Ui is closed. Moreover, H ∩ Ui = Hi is open in H. The following example is attributed by J. Dixon in [28, 29] to H. Salzmann. Example 2.23. Let S3 = ℤ(3) ⋊ ℤ(2) and let G be the subgroup of P := (S3 )ℕ whose elements (xn )n∈ℕ have all but finitely many components equal to the identity and equip G with the discrete topology. Let N ⊆ G be the subgroup of all (xn )n∈ℕ such that xn = (an , 0) with an ∈ ℤ(3) for all n. We consider two 2-Sylow subgroups S = {0} × ℤ(2) and T = {(0, 0), (1 + 3ℤ, 1 + 2ℤ)} of S3 and set H1 := G ∩ Sℕ

and H2 := G ∩ T ℕ .

Then both H1 and H2 are complements for N in G. If H1 = H2g for some g = (gn )n∈ℕ ∈ P, then gn ≠ 1 must hold for all n ∈ ℕ. However, such an element g cannot belong to G. This example shows that the conjugacy conclusion may fail if G/N is not compact. An example of good conjugacy is the following. Proposition 2.24. Let G be a periodic group, σ a set of primes, and N a closed normal σ-subgroup. If G/N is sigma-compact and compactly ruled, then all σ-Sylow subgroups are conjugate. In particular, N is contained in every one of them. Proof. Let H and K both be σ-Sylow subgroups of G. By Proposition 2.12 (ii), both of them contain N and by Proposition 2.12 (iii), both quotient groups H/N and K/N are σ-Sylow subgroups of G/N. By the Schur–Zassenhaus Theorem 2.21 (ii) there exists g ∈

2.3 A Schur–Zassenhaus theorem

| 29

G such that (gN)K(gN)−1 /N ⊆ H/N. Since both of these groups are Sylow subgroups, equality holds. This equality translates into K = KN = gHg −1 N = gHg −1 , proving the desired conjugacy statement. Since N is contained in at least one σ-Sylow subgroup by Lemma 2.4 and all of them are conjugate, the normal subgroup N is contained in each one of them. Whatever result we may envisage of the type that a normal Sylow subgroup N in a periodic locally compact group G has a complement H provided it satisfies some additional conditions, it will necessarily entail for the complement H the property that it is a σ 󸀠 -Sylow subgroup. This motivates here the question what kind of situation we encounter if we face a normal σ-Sylow subgroup N together with a σ 󸀠 -Sylow subgroup H. The following result uses the Schur–Zassenhaus theorem, but it is otherwise rather elementary. It does provide an instance of the Mayer–Vietoris formalism, Proposition 1.31, that is based on algebraic data and not on sigma-compactness arguments. Theorem 2.25. Let N be a normal σ-Sylow subgroup of a locally compact periodic group G. Then a σ 󸀠 -Sylow subgroup H of G exists such that NH is an open and hence closed subgroup. Moreover, if H is any σ 󸀠 -Sylow subgroup of G, then NH is closed in G and H is a complement of N in NH, that is, NH = N ⋊ H. Proof. Let U be an open subgroup of G containing N so that U/N is compact; such a group exists since G/N, as a periodic locally compact group, is totally disconnected. Theorem 2.21 applies to U and shows that U = N ⋊ CU with a compact σ 󸀠 -Sylow subgroup CU of U. Then by Lemma 2.4 there is a σ 󸀠 -Sylow subgroup H of G containing CU , and H is closed by Corollary 2.7. Since the product NH contains the open subgroup U = NCU , it is open and therefore closed itself. Now let H be an arbitrary σ 󸀠 -Sylow subgroup. By Corollary 2.7 again, H is a closed subgroup. This yields the locally compact (external) semidirect product N ⋊α H with respect to the action of H by inner automorphisms on N and gives an injective morphism of locally compact groups μ: N ⋊α H → G, μ(n, h) = nh onto the subgroup NH of G. Claim (i): NH is closed in G. Proof of the claim: Since N ⊆ U, the modular law shows that NH ∩ U = N(U ∩ H). Hence by Theorem 2.21 there is an element u ∈ U such that u−1 (H ∩ U)u ⊆ CU , that is, H ∩ U ⊆ uCU u−1 , and since uCU u−1 is a complement of N in U, there is no harm in assuming u = 1. Thus H ∩ U ⊆ CU . By Theorem 2.21, the restriction and corestriction μ|(N ⋊α CU ) : N ⋊α CU → U is an isomorphism of locally compact groups, and thus μ|(N ⋊α (H ∩ U)) : N ⋊ (H ∩ U) → NH ∩ U

30 | 2 Periodic locally compact groups and their Sylow theory is an isomorphism of locally compact groups. The compact group H ∩ U is closed in CU and so N × (H ∩ U) is closed in N × CU and therefore NH ∩ U is closed in U. Since U is open, the subgroup NH is locally closed in G and is therefore closed in G. Claim (ii): The corestriction μ: N ⋊ H → NH is an isomorphism of locally compact groups. For a proof of this claim, after (i), there is no harm in writing NH = G in order to simplify the notation. We know that N(H ∩U) ⊆ NCU = U. Now let u ∈ U. From G = NH there exist elements n ∈ N and h ∈ H such that u = nh. Then h = n−1 u ∈ H ∩ U, whence u = nh ∈ N(H ∩ U). Thus we have the equality U = N(H ∩ U). From (i) we know that H ∩ U ⊆ CU and N ⋊ CU ≅ NCU = U; moreover, N ⋊α (H ∩ U) ≅ N(H ∩ U) = U. This allows us to conclude CU = H ∩ U. Therefore, from (i) again, H ∩ U is a complement of N in U. We have the following commuting diagram:

The vertical maps are embeddings. The top horizontal map is an isomorphism of locally compact groups. The locally compact subgroups U ∩ H and U are open in the groups H and G, respectively, and so the groups in the upper line are open in their lower line counterparts. Thus the morphism μ is locally open at 1 and therefore is open and so is an isomorphism of locally compact groups. The remarkable weakness of the hypotheses of Theorem 2.25 is something noteworthy. The conclusion NH = N ⋊ H is obtained without any assumption of sigmacompactness. However, the possibility that for a normal σ-Sylow subgroup N and for a σ 󸀠 -Sylow subgroup the subgroup NH of G may be proper occurs already in the discrete case of a locally finite group as the following example shows. An exponent 2 group is always abelian and is algebraically a vector space over the field with two elements. Example 2.26. An example of Kovács, Neumann, and de Vries displays in [30, p. 68] a locally finite group G which is a nonsplit extension of a group of exponent 3 by a group of exponent 2. The group is a subgroup of a power of the six-element group S3 . So Theorem 2.25 cannot be improved by asserting that N must have a complement. The example also illustrates the significance of the hypothesis in Theorem 2.21 saying that G/N should be sigma-compact. The full power of Theorem 2.25 still is attained of course if G = NH. If one defines 𝒰 to denote the set of all open subgroups U containing N such that U/N is compact and assumes that G/N is compactly ruled (which is certainly the case if G itself is compactly

2.3 A Schur–Zassenhaus theorem

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ruled by Proposition 1.9), then G = ⋃U∈𝒰 U. Alternatively, this means that G is the direct limit of the open subgroups U of 𝒰 . Then G = NH ⇔ (∀U ∈ 𝒰 ) U = (NH ∩ U) = N(H ∩ U).

(∗)

In the following we have a special instance where this yields G = NH. We shall see that the centrality of a Sylow subgroup in Corollary 2.27 will yield a first example of a normal Sylow subgroup having a complement by enforcing G = NH through additional hypotheses. The Černikov Theorem 2.31 is another such instance. Corollary 2.27. Let N be a central σ-Sylow subgroup of a periodic locally compact group and assume that G/N is compactly ruled. Then N has a unique complement H = Gσ󸀠 and G is the direct product N × H. Proof. We define 𝒰 as in the preceding paragraph. Since N is central, in each U the central subgroup U ∩ N = Uπ , π = π(N), has a unique complement CU = Uπ 󸀠 and U = Uπ × Uσ󸀠 . The subset H := Gσ󸀠 is closed by Lemma 2.6. Since U ∩ Gσ󸀠 = Uσ󸀠 , we have CU = H ∩ U. The relation G = ⋃U∈𝒰 U entails H = ⋃U∈𝒰 CU and thus H is a subgroup, evidently the unique σ 󸀠 -Sylow subgroup of G, and (∗) shows that G = NH = N × H. Since H is normal, we have the relation H = Gσ󸀠 from Proposition 2.18. For σ = {p} we obtain the following. Corollary 2.28. Let N be a central p-Sylow subgroup of a periodic locally compact group G and let G/N be compactly ruled. Then N = Gp , and Gp󸀠 is a p󸀠 -Sylow subgroup such that G = Gp × Gp󸀠 . The following proposition, although being somewhat technical, fits right into the subject of normal Sylow subgroups. Proposition 2.29. Let G be a periodic group and N a normal σ-Sylow subgroup of G. Then we have the following conclusions: (a) for every subgroup S of G the intersection S ∩ N is a σ-Sylow subgroup of S, (b) let B be a closed σ 󸀠 -subgroup; then NB is closed in G. If, in addition, CG (N)/Z(N) is compactly ruled, then the following statements hold: (c) if S = CG (N), then B = Sσ󸀠 is a normal complement of S ∩ N in S and S is the direct product (S ∩ N) × B and is a closed subgroup of S, (d) L := NB = N ⋊ B is a closed normal subgroup of G. Proof. (a) By Proposition 2.18 we have N = Gσ and we have to show S ∩ N = Sσ ; but this follows at once from Sσ = S ∩ Gσ . (b) Lemma 2.4 implies the existence of a σ 󸀠 -Sylow subgroup H of G containing B. Theorem 2.25 implies that NH ≅ N ⋊H is a closed subgroup of G. Since, as a topological

32 | 2 Periodic locally compact groups and their Sylow theory space, this subgroup is homeomorphic to the product space N × H it follows at once that N × B is a closed subspace and thus NB ≅ N ⋊ B is a closed subgroup of G. (c) Now let S = CG (N). Then [N, S] = {1} and so [S ∩ N, S] ⊆ [N, S] = {1}. Hence S∩N = Z(N) is a central σ-Sylow subgroup of S. Then Corollary 2.27 yields S = (S∩N)×B for the characteristic subgroup B of the normal subgroup S of G. Note that S ∩ N is also central in N and thus in NS = NB. (d) The subgroup NB = N ⋊B is closed in G by (b) and equals NS = N ×B by (c). Equipped with these results we prove a locally compact version of a result of N. S. Černikov (see [19]). Proposition 2.30. Let G be a periodic group with a normal σ-Sylow subgroup N. Suppose that the factor group G/NCG (N) is compact and CG (N)/Z(N) is compactly ruled. Then the following statements hold: (i) N has a complement in G; and (ii) every two complements of N in G are conjugate. Proof. Put L := NCG (N). Proposition 2.29 (d) shows that L is closed in G and is topologically the direct product L = N × Lσ󸀠 . G/L 󸀠 Next observe that the compactness of G/L implies that L/L σ󸀠 is compact and, thus, σ

Theorem 2.21 yields a complementary σ 󸀠 -subgroup in G/Lσ󸀠 for L/Lσ󸀠 . Its preimage in G we call S. Then G = NS and Lemma 2.19 shows that S is a σ 󸀠 -Sylow-group. For proving that S is a complement we apply Theorem 2.25 with S playing the role of H there. (ii) Let H and T be complements of N in G. Then Hσ󸀠 ⊆ T since Hσ󸀠 is the unique 󸀠 σ -Sylow subgroup of H = NCG (N) = N × Hσ󸀠 . Then, modulo Hσ󸀠 we have G/Hσ󸀠 = N ⋊ S = N ⋊ T

and thus the Schur–Zassenhaus Theorem 2.21 provides a g ∈ G with T ⊆ Sg Hσ󸀠 . Therefore T = Sg . Theorem 2.31. Let G be a periodic group with a normal σ-Sylow subgroup N and CG (N)/Z(N) compactly ruled. Then L := NCG (N) is a closed normal subgroup of G and we assume that G/L is sigma-compact. Then N has a complement in G, and every complement is a σ 󸀠 -Sylow subgroup of G. Proof. There is an ascending chain of open subgroups Gi each containing L with G = ⋃∞ i=1 Gi . For every i use Proposition 2.30 in order to decompose Gi = Ni ⋊ Si . Certainly Si+1 ∩ Gi is a complement for N in Gi . Therefore, by the second part of that proposition gi for every i there is gi ∈ Gi with Si ⊆ Si+1 . Using this fact, one can inductively achieve that also the groups Si form an ascending chain of subgroups of G. Let S be their union.

2.4 The fixed point theorem

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Then G = N ⋊ S. The subgroup S is closed since Si = S ∩ Gi is closed in the open closed subgroup Gi . In order to note that every complement is a σ 󸀠 -group we refer to Lemma 2.19.

2.4 The fixed point theorem The Schur–Zassenhaus theorem deals with the theme of a normal σ-Sylow subgroup N of a periodic group G with a closed normal subgroup N. It firstly asserts the existence of a complement H of N and, secondly, discusses the issue of the conjugacy of the complements of N. Here we address a fixed point theorem pertinent to the second of these issues. For profinite groups it may be found in [93], Proposition 2.3.16. We shall prove the theorem for totally disconnected compactly ruled groups—a vastly larger class. Recall Definition 2.14 of an automorphic action and the definition of the inner automorphism Ig sending every x ∈ G to Ig (x) := gxg −1 . If Ω is any subgroup of G, we say that a subgroup H of K is Ω-invariant iff Iω (H) = H, that is, ωHω−1 = H for all ω ∈ Ω in G. Lemma 2.32. Let Γ act automorphically on a group K and let H be a Γ-invariant subgroup of K. Let G = K ⋊ Γ be as in Remark 2.15. For an element k ∈ K the following statements are equivalent: (1) the coset kH is Γ-invariant, (2) H is k −1 Γk-invariant in G, (3) H is ⟨Γ ∪ k −1 Γk⟩-invariant in G. Proof. Since H is Γ-invariant, clearly (2) and (3) are equivalent. We have to show that (1) and (2) are equivalent. We shall use the identification of K and Γ with the subgroups of K × {1}, respectively, {1} × Γ of G. Then (1) is equivalent to (∀γ) kH = Iγ (kH) = γkHγ −1 = γkγ −1 H, that is, (∀γ) [k −1 , γ] ∈ H.

(∗)

Since H is Γ-invariant we have γ −1 Hγ = H and so (∗) implies H = [k −1 , γ]H[γ, k −1 ] = Ik−1 (γ)γ −1 HγIk−1 (γ)−1 = II −1 (γ) (H) for all γ ∈ Γ and that is condition (2). Conversely, if k

(2) holds, then (∀γ) γkγ −1 H = γ(kH)γ −1 = kH, and this is equivalent to [k −1 , γ] ∈ H for all γ which is (∗) and therefore equivalent to (1). This completes the proof of the lemma.

Lemma 2.33. Assume that K and Γ are compactly ruled locally compact groups and Γ acts automorphically on K. Then the following statements hold: (a) G = K ⋊ Γ is compactly ruled, (b) if k ∈ K and Δ ⊆ Γ is a compact subgroup, then ⟨Δ ∪ k −1 Δk⟩ is a compact subgroup Δk of G acting automorphically on K by inner automorphisms of G,

34 | 2 Periodic locally compact groups and their Sylow theory (c) if H is a closed Δ-invariant subgroup of K, then the coset kH is Δ-invariant if and only if H is Δk -invariant, (d) if H is a closed subgroup such that H and the coset kH are both Δ-invariant, then there is a compact open subgroup U of K containing k such that U is Δk -invariant, (e) in particular, U ⋊ Δ is a compact group containing k, U, and Δ, in which both U ∩ H and k(U ∩ H) are Δ-invariant. Proof. (a) Every compact subset of G is contained in one of the form C × Δ, and since both K and Γ are compactly ruled groups, we may assume that C is a subgroup of K and Δ is a subgroup of Γ. Then the subspace Δ⋅C ⊆ K is compact, whence ⟨Δ⋅C⟩ is a compact subgroup of K since K is compactly ruled; it is also Δ-invariant, whence ⟨Δ⋅C⟩ × Δ is a compact subgroup of G containing C × Δ. Thus G is compactly ruled as well. (b) By (a), the subgroup Δk of G is compact, and since K is normal in G Δk it acts automorphically on K by inner automorphisms. (c) This is a consequence of Lemma 2.32. (d) We know that H is Δk -invariant. As subgroup of K, the group H is a compactly ruled group and thus has a compact open subgroup V containing k. Then Δk ⋅V is compact and contained in H. Since H is compactly ruled, U = ⟨Δk ⋅V⟩ is a compact subgroup of H which, in addition, is Δk -invariant and contains k. (e) Now U ⋊ Δ is a compact subgroup of G containing k, U, and Δ. Also, U and H are Δk -invariant, whence U ∩ H is Δk -invariant. In particular U ∩ H is both Δ- and k −1 Δk-invariant and thus Lemma 2.32 applies to show that k(U ∩ H) is Δ-invariant. Notation 2.34. Let G be a group and Γ a subgroup of its automorphism group. For any subset X ⊆ G we let CX (Γ) denote the elements in X fixed by every automorphism γ ∈ Γ. If Γ is generated by γ we shall also write CG (γ) in place of CG (⟨γ⟩). Lemma 2.35. Assume that K and Γ are totally disconnected compactly ruled locally compact groups and Γ acts automorphically on K. Let H ⊆ K be a closed subgroup and k an element of K such that both H and kH are Γ-invariant. Assume in addition that π(H) ∩ π(Γ) = 0.

(†)

Now let Δ be any compact open subgroup of Γ. Then CkH (Δ) ≠ 0. Proof. Lemma 2.33 applies in its entirety and we find a profinite open subgroup U of K such that U ∩ H and k(U ∩ H) are Δ-invariant. Then U ⋊ Δ is a profinite group satisfying π(U) ∩ π(Δ) = 0. Then [36, Lemma 1.3] applies to provide the claim. In the group G = K ⋊ Γ with the condition (†) this means exactly the Schur– Zassenhaus condition of Definition 2.20 for the normal subgroup N = K × {1}. Consider for a moment a group Γ acting automorphically on a group K containing a Γ-invariant subgroup H and an element f fixed by Γ. Then fH is obviously a Γ-invariant coset. The following fixed point theorem provides a converse provided sufficient hypotheses are satisfied.

2.4 The fixed point theorem

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Theorem 2.36. Let K be a totally disconnected locally compact compactly ruled group on which a totally disconnected locally compact compactly ruled group Γ acts automorphically and let H ⊆ K be Γ-invariant. Assume the following conditions: (a) π(H) ∩ π(Γ) = 0; (b) there is a k ∈ K such that the coset kH is Γ-invariant. Then CkH (Γ) ≠ 0. Remark 2.37. The conclusion says that any Γ-invariant coset kH of K contains a Γ-fixed point. Proof. We shall consider G = K ⋊ Γ as in the preceding lemmas. By Lemma 2.35, for every profinite compact open subgroup Δ of Γ, we have CkH (Δ) ≠ 0. Let 𝒰 denote the covering of Γ by profinite open subgroups. By Proposition 1.3 this covering is up-directed with respect to containment. If Δ ⊆ Ω in 𝒰 , then CK (Ω) ⊆ CK (Δ). Thus {CkH (Δ) : Δ ∈ 𝒰 } is a filter basis of compact subsets of kH and hence has a nonempty intersection. Since Γ = ⋃ 𝒰 , an element in this intersection is a fixed point of Γ. This proves CkH (Γ) ≠ 0. Remark 2.38. Assume K and Γ are totally disconnected locally compact compactly ruled groups where Γ acts automorphically on K leaving a closed subgroup H of K invariant. Let K/H = {xH : x ∈ K} denote the homogeneous space of left-cosets. Then Γ acts continuously on K/H via γ⋅(xH) = (γ⋅x)H. Now the following is just a re-wording of Theorem 2.36. Corollary 2.39. Under the conditions of Remark 2.38 assume in addition that π(H) ∩ π(Γ) = 0. Then CK/H (Γ) = CK (Γ)⋅H. Let us re-formulate the fixed point theorem in our earlier terminology referring to the Schur–Zassenhaus condition. Let G be a topological group and M ⊆ N closed normal subgroups of G and let H be a complement of N so that in fact G = N ⋊ H. Then H acts on N under inner automorphisms and on N/M in the obvious fashion via h⋅(nM) = hnh−1 M. In this terminology the fixed point theorem reads as follows. Theorem 2.40. Let G be a periodic compactly ruled group of the form G = N ⋊ H and suppose that N satisfies the following conditions: (i) N satisfies the Schur–Zassenhaus condition, (ii) N contains a closed H-invariant subgroup M. Then CN/M (H) =

CN (H)M . M

36 | 2 Periodic locally compact groups and their Sylow theory If the subgroup N is abelian we have some special variations of this theme. In view of these combinatorial facts, the next corollary now follows with Γ = G/CG (K) immediately from Corollary 2.39. Corollary 2.41. Let G be a totally disconnected compactly ruled group with a normal subgroup N and suppose that the following conditions are satisfied: (i) N satisfies the Schur–Zassenhaus condition, (ii) N is abelian. Then N = (Z(G) ∩ N)[N, G]. Proof. By (ii), the factor group Γ := G/N acts on N via gN⋅k = gkg −1 , and CN (Γ) = CN (G) = Z(G) ∩ N.

(1)

Furthermore, since N is normal, it is Γ-invariant. We set M = [G, N] and note that M is normal in G. Now Γ also acts on N/M via (gN)⋅(kM) = gkg −1 M. For all k ∈ N we have [k, G] ⊆ [N, G] and so (∀g ∈ G, k ∈ N) gkg −1 ∈ kM, and that means (∀k ∈ N) kM ∈ CN/M (Γ). Therefore N/M = CN/M (Γ).

(2)

Now we apply Corollary 2.39, using (1), and derive from (2) the equation N/M = CN (Γ)⋅M =

(Z(G) ∩ N)[N, G] . M

(3)

Thus N = (Z(G) ∩ N)[N, G]. The Schur–Zassenhaus condition played its role in Theorems 2.21, 2.25, 2.28, and 2.31 and will play a role for Theorems 2.44, 2.53, and 2.36. The following result, which relates to findings of the second author in [51] and also to ones by Čarin in [18], fits into the series. Proposition 2.42. Let G be a periodic compactly ruled group G with a closed normal Sylow subgroup N and a complement H satisfying the following hypotheses: (1) N satisfies the Schur–Zassenhaus condition, (2) N is abelian, (3) N contains arbitrarily small compact subgroups open in N, (4) At least one of Z(G) ∩ N and [N, H] is sigma-compact. Then the following statements hold: (i) [H, [H, N]] = [H, N], (ii) N = (Z(G) ∩ N) × [H, N].

2.4 The fixed point theorem

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Proof. For proving (i) it suffices to remark that as a consequence of the commutator identity [a, bc] = [a, b][b, [a, c]][a, c] one arrives at [H, N] = [H, CN (H)[H, N]] = [H, [H, N]]. For proving (ii) we note from Corollary 2.41 that N = (Z(G) ∩ N)[H, N]. By Proposition 1.27 and Condition (4) we know that there is a quotient morphism (Z(G) ∩ N) × [N, H] → N with kernel (Z(G) ∩ N) ∩ [N, H]. Thus the rest of the proof must be devoted to showing that Z(G) ∩ N intersects [H, N] trivially. Let x ∈ CN (H) ∩ [H, N] and suppose that x ≠ 0. By (3) there is a compact open subgroup U of N which is normal in G and does not contain x. We must derive a contradiction. Now G/U = N/U ⋊ H contains the discrete abelian subgroup N/U. It now will be enough to prove the contradiction under the additional assumption N being a discrete subgroup of G. This means that there are elements ni ∈ N and hi ∈ H, and k ∈ ℕ such that x = ∑ki=1 (hi − 1)⋅ni = 0 whenever (h − 1)⋅x = 0 for all h ∈ H. Since G is compactly ruled the finitely generated subgroup Δ := ⟨ni | i = 1, . . . , k⟩ is profinite and hence Nk := N ∩ Δ is a finite abelian group on which Hk := ⟨h1 , . . . , hk ⟩ acts. Since Nk is finite, the kernel Kk of the action of Hk on Nk is open in Hk and hence we have a coprime action of the finite group Γk := Hk /Kk on Ak . It follows from Maschke’s theorem (see [60, p. 118, 17.6 Satz]) that CNk (Hk ) has a Hk -invariant complement in Nk , i. e., Nk = CNk (Hk ) × B. Let us show that B = [Hk , A]. Observe first that [Hk , Nk ] = [Hk , CNk (Hk ), B] = [Hk , B] and since B is Hk -invariant, we find that [Hk , B] is contained in B. If B/[Hk , B] is not trivial, then, using Corollary 2.39, one can lift fixed points and thus CB (Γk ) is a nontrivial subgroup of B. However CB (Hk ) is also a subgroup of CN (Hk ), contradicting

38 | 2 Periodic locally compact groups and their Sylow theory CN (H) ∩ B = {0}. Thus we have B = [Hk , B] and from the above B = [Hk , N]. We have now the decomposition Nk = CNk (Nk ) × [Nk , Hk ], from which the desired contradiction, x = 0, follows. Thus CG (H) ∩ [H, N] = {0}, and this is what we still had to show.

2.5 Pairwise commuting p-Sylow subgroups In Section 2.2 we maintained the normality of a Sylow subgroup and used the Schur– Zassenhaus condition in most of what was deployed. If N is a normal p-Sylow subgroup, we know NS = SN for all q-Sylow subgroups S. Now we strengthen the condition of normality to the condition that [N, S] = {1} for all q ≠ p. We observe in passing that this condition is weaker than the centrality of N which we already encountered in Corollaries 2.27 and 2.28. Lemma 2.43. Assume that for the compactly ruled group G and for a prime p we have [Gp , Gp󸀠 ] = {1}.

(#)

Set N := ⟨Gp ⟩ and H := ⟨Gp󸀠 ⟩. Then N and H are closed normal subgroups and the following conclusions hold: (i) [N, H] = {1}, (ii) NH = G, (iii) N ∩ H ⊆ Z(G), where Z(G) is the center of G, (iv) let S be a p-Sylow subgroup; then G = S × H. Proof. Since both N and H are characteristic, they are normal. (i) Since the centralizer of a set is a subgroup, the relation Gp󸀠 ⊆ CG (Gp ) implies H ⊆ CG (Gp ), which is equivalent to Gp ⊆ CG (H). Thus N ⊆ Gp , and so (i) follows. (ii) By (i) NH is a subgroup and by the definitions of N and H it contains all q-Sylow subgroups q ∈ π(G). Then Lemma 2.8 shows that it is G. (iii) By (i) N ∩ H commutes elementwise with both N and H. Then by (ii) it commutes with G and thus is central. (iv) Since S ⊆ N, we have [S, H] = {1} by (i). We claim that S ⊈ H. So suppose, by way of contradiction, that S ⊆ H. Then S ⊆ N ∩ H ⊆ Z(G) by (iii). So G = S × Gp󸀠 by Corollary 2.28. Since Gp󸀠 is a subgroup in this case, it follows that Gp󸀠 = H yielding S ∩ H = {1}, a contradiction. We now assume S ⊈ H and consider G∗ := G/(N ∩ H), N ∗ := N/(N ∩ H), and ∗ H := H/(N ∩ H). Then G∗ = N ∗ H ∗ and N ∗ ∩ H ∗ = {1}. Any q-element gH ∈ G/H with q ∈ p󸀠 is the image of a q-element in G, say g; but then g ∈ H and therefore gH = H.

2.5 Pairwise commuting p-Sylow subgroups | 39

Thus G/H is a p-group and thus N ∗ , which maps bijectively onto NH/H = G/H, is a p-group. The central subgroup N ∩ H of N, as an abelian periodic group, is of the form (N ∩ H)p × Z with a central p󸀠 -group Z of N. (See, e. g., Theorem 3.3.) Since N/Z is a p-group, Z is necessarily a central p󸀠 -Sylow subgroup of N. Thus by Corollary 2.28, there is a p-Sylow subgroup S of N such that N = S × Z. By definition of N and H we have S ⊆ N and Z ⊆ H. Thus G = NH = SH, whence G = S × H by Theorem 2.25. A projection is a morphism pr: G → H for a subgroup H ∈ G such that pr |H = idH . Recall that for a prime p we let p󸀠 denote the set theoretic complement in π(G). Let us summarize what we have. Theorem 2.44. Let G be a compactly ruled group, p a prime number. Then the following conditions are equivalent: (1) [Gp , Gp󸀠 ] = {1}, (2) Gp and Gp󸀠 are subgroups and G = Gp × Gp󸀠 , (3) there is a unique projection prp : G → Gp with kernel Gp󸀠 . Proof. Lemma 2.43 shows (1)⇒(2). The implication (2)⇒(1) is trivial. Likewise, the equivalence (2)⇔(3) is immediate. Both Gp and Gp󸀠 are normal Sylow subgroups satisfying the Schur–Zassenhaus condition. The primes p occurring in Theorem 2.44 obviously are rather special, and it is practical to give them a name of their own. Definition 2.45. For a periodic locally compact group G we set ν(G) = {p ∈ π(G) : [Gp , Gp󸀠 ] = {1}}. Lemma 2.46. Let G be a compactly ruled group. Assume that ν(G) is not the empty set. Then P := Gν(G) Gν(G)󸀠 is a direct product and G = Gν(G) × Gν(G)󸀠 .

(1)

Proof. For p ∈ ν(G), every p-Sylow subgroup agrees with Gp and commutes elementwise with all other Sylow subgroups. By Theorem 2.44 (3), for each prime p ∈ ν(G) we then have a projection prp : G → Gp with kernel Gp󸀠 . This gives a canonical morphism θ: G → ∏ Gp , p∈ν(G)

θ(g) = (prp (g))p∈ν(G) .

We observe ker θ = ⋂ ker prp󸀠 = ⋂ Gp󸀠 = Gν(G)󸀠 . p∈ν(G)

p∈ν(G)

40 | 2 Periodic locally compact groups and their Sylow theory Now we apply Theorem 2.25 with N = Gν(G)󸀠 and H = Gν(G) , which is normal in this case, and so we know that P := Gν(G) Gν󸀠 (G) is closed in G and is a direct product P ≅ Gν(G) × Gν(G)󸀠 . However, for any p ∈ π(G) = ν(G) ∪ ν(G)󸀠 any p-Sylow subgroup S is contained in Gν(G) or in Gν(G)󸀠 . Thus P contains all p-Sylow subgroups of G. Then by Lemma 2.8, P = G, that is, G = Gν(G) × Gν(G)󸀠 .

(1)

This completes the proof of the lemma. We now determine the structure of Gν(G) . In order to simplify the notation, we assume G = Gν(G) ; then ker θ is singleton and so θ is injective. Lemma 2.47. If G = Gν(G) , then θ(G) = ∏ Gp .

(2)

p∈π(G)

Proof. For an element g ∈ G we use the notation gp = prp (g) ∈ Gp ⊆ G and thus θ(g) = (gp )p∈π(G) ∈ ∏p∈π(G) Gp . Now, for each finite subset F ∈ π(G), Theorem 2.44 allows us by induction to define an isomorphism θF : GF → ∏p∈F Gp . Accordingly, if ∏p∈F Gp is naturally identified with the obvious partial product PF of ∏p∈π(G) Gp , then we know (∀F finite ⊆ π(G)) PF ⊆ θ(G) ⊆ ∏ Gp . p

Equivalently, ⋃

F finite ⊆π(G)

PF ⊆ θ(G) ⊆ ∏ Gp p

and as a consequence, θ(G) is dense in the product, that is, θ(G) = ∏ Gp , p∈π(G)

(2)

as is asserted in the lemma. This is as close as we can get to surjectivity with θ. Now let U be any compact open subgroup of G and p ∈ π(G). Since Gp is the set of all p-elements of G, Up = U ∩ Gp is the set of all p-elements of U, and since U ∩ Gp is a subgroup of U, we know that Up is the unique p-Sylow subgroup of U. We can now apply Lemma 2.47 to U in place of G and conclude θ(U) = ∏p∈π(U) Up . However,

2.5 Pairwise commuting p-Sylow subgroups | 41

since U and therefore θ(U) are compact, we have θ(U) = ∏p Up . More specifically, θ|U : U → ∏p Up is an isomorphism of compact groups and we have the following commutative diagram of morphisms of topological groups

incl

θ|U

U ↑ ↑ ↑ ↑ ↓ G

󳨀󳨀󳨀󳨀→ 󳨀󳨀󳨀󳨀→ θ

∏p Up ↑ ↑ ↑ ↑ incl ↓ ∏ p Gp .

All groups in the diagram are locally compact with the possible exception of the product ∏p Gp , which is locally compact iff the number of noncompact factors is finite. The proof of the following observation is an exercise. Remark 2.48. Let Gp , p ∈ π be a family of locally compact p-groups with a family of compact open subgroups Up . Then the set DU of all (gp )p ∈ ∏p Gp such that {p ∈ π : gp ∉ Up } is finite is a dense subgroup of ∏p Gp containing ∏p Up . Definition 2.49. Let (Gj )j∈J be a family of locally compact groups and assume that for each j ∈ J the group Gj contains a compact open subgroup Cj . Let P be the subgroup of the cartesian product of the Gj containing exactly those J-tuples (gj )j∈J of elements gj ∈ Gj for which the set {j ∈ J : gj ∉ Cj } is finite. Then P contains the cartesian product C := ∏j∈J Cj which is a compact topological group with respect to the Tychonoff topology. The group P has a unique group topology with respect to which C is an open subgroup. Now the local product of the family ((Gj , Cj ))j∈J is the group P with this topology, and it is denoted by loc

P = ∏(Gj , Cj ). j∈J

Before continuing let us introduce a notation for the local product in which all factors are equal. Definition 2.50. Consider a pair (A, B) of topological groups such that B is a compact open normal subgroup of A. The local product ∏loc j∈J (Aj , Bj ), where Aj = A and Bj = B for all j ∈ J, we shall write

as (A, B)loc,J and call it a local power.

We observe (A, A)loc,J = AJ and (A, {0})loc,J = A(J) . Definition 2.51. In the circumstances of Remark 2.48, the dense subgroup D of the full product of the Gp will be denoted (Gp , Up ). ∏dense p Lemma 2.52. Let U be a compact open subgroup of the periodic group G satisfying (Gp , Up ). π(G) = ν(G). Then the image of θ: G → ∏p Gp is ∏dense p

42 | 2 Periodic locally compact groups and their Sylow theory Proof. We saw that θ|U implements an isomorphism U ≅ ∏p Up . Now let g be an arbitrary element of G. Then ⟨g⟩ is a profinite subgroup since G is periodic, and θ(⟨g⟩) = ⟨(gp )p ⟩ = ∏p ⟨g⟩p . The open subgroup ⟨g⟩∩U = ∏p (⟨g⟩p ∩Up ) of ⟨g⟩ has finite index in ⟨g⟩ = ∏p ⟨g⟩p . This index is ∏p (⟨gp ⟩ : (⟨gp ⟩ ∩ Up )) and therefore (⟨g⟩p : (⟨gp ⟩ ∩ Up )) = 1 for all p ∈ π(G) with the exception of the primes p from some finite subset Fg ⊆ π(G). This means that for all p ∈ π(G) \ Fg we have gp ∈ Up . Conversely, let (gp )p ∈ ∏p Gp be an element with gp ∈ Up for all p ∈ π(G) \ F for some finite set F of primes. Write F = {p1 , . . . , pn } in ascending order and set h := gp1 ⋅ ⋅ ⋅ gpn ∈ G. We notice that θ(h) = (hp )p∈π(G) ,

gp , if p ∈ F, where hp = { 1, otherwise.

For each p ∈ π(G) we define 1, if p ∈ F, up = { gp , otherwise. Since θ|U: U → θ(U) is an isomorphism, there is an element u ∈ U such that θ(u) = (up )p∈π(G) . We observe directly that θ(hu) = θ(h)θ(u) = (hp up )p = (gp )p . This shows that θ(G) = ∏dense (Gp , Up ). p Notice that the conclusion of this lemma does not depend on the choice of the compact open subgroup U. Since the subgroup θ(G) is dense, but is a proper subgroup of ∏p Gp , it will often fail to be locally compact. Indeed θ(G) is locally compact iff θ is bijective iff all but finitely many of the factors Gp are compact. The local product is always locally compact. It is clear that the morphism loc

θ: G → ∏(Gp , Up ) p

is bijective. Adopting this terminology, we can now summarize our results on periodic locally compact groups, Such groups always have a compact open subgroup as they are totally disconnected locally compact. We recall ν(G) ⊆ π(G) from Definition 2.45. Theorem 2.53. In a compactly ruled locally compact group G, the set Gν(G)󸀠 of α-elements with α ∩ ν(G) = 0 is a closed normal subgroup, and all p-Sylow subgroups

2.6 Sylow bases in inductively prosolvable groups | 43

for p ∈ ν(G) are normal subgroups. Moreover, loc

G ≅ Gν(G)󸀠 × ∏ (Gp , Up ) p∈ν(G)

for a suitable family of compact open subgroups Up ⊆ Gp as p ranges through ν(G). In particular Gν(G) is a local product ∏loc p∈π(G) (Gp , Up ) of its normal p-Sylow subgroups for any compact open subgroup U of Gν(G) . The following definition therefore appears to be appropriate. Definition 2.54. In a compactly ruled locally compact group, the canonical closed normal subgroup Gν(G) will be called the nilfactor of G. Definition 2.55. For a periodic locally compact group G let ζ (G) be the subset of all p in ν(G) with Gp being an abelian subgroup of G. Put Gζ (G) = ⟨Gp | p ∈ ζ (G)⟩. The local product structure of Gν(G) allows us to conclude. Corollary 2.56. Let G be a periodic compactly ruled group. Then G = Gζ (G) × Gζ (G)󸀠 = Gζ (G) × Gν(G)\ζ (G) × Gν(G)󸀠 . The sets ν(G) and ζ (G) will play significant roles in later parts of the text, where we discuss the Sylow subgroup structure of near abelian groups in greater detail.

2.6 Sylow bases in inductively prosolvable groups A discrete group G is locally 𝒫 , for a property 𝒫 , if every finitely generated subgroup enjoys 𝒫 . For locally compact groups an analogous notion will be considered. Notation 2.57. We say that H is finitely generated iff there is a finite subset F ⊆ H such that H = ⟨F⟩. Definition 2.58. For a property 𝒫 and locally compact group G we say that G is inductively 𝒫 provided every finitely generated subgroup satisfies 𝒫 . Remark 2.59. In Chapter 5, we shall encounter groups which are “inductively monothetic”. Let G be a periodic locally compact group, which is compactly ruled (see Definition 1.4), i. e., “inductively finitely generated”. Now we assume that G is inductively 𝒫 . Let U be a compact (open) subgroup. As a profinite group, U has quotient morphisms f : U → E onto finite groups. For each of these there is a finite subset F ⊆ U with f (F) = E. Let H = ⟨F⟩. Then H is a finitely generated closed subgroup and E = f (H). Further H has property 𝒫 by assumption. So U is pro-𝒫 .

44 | 2 Periodic locally compact groups and their Sylow theory The projective limit of finite solvable groups is called prosolvable. See [93, 113]. We alert the reader that we use the term prosolvable expressively for profinite groups. For pro-Lie groups a considerably larger class of groups is called prosolvable. (Cf. [52].) Thus G is inductively prosolvable provided every finitely generated subgroup is prosolvable. Definition 2.60. A Sylow base of a periodic group is a selection of p-Sylow subgroups Sp , one for each p ∈ π(G), such that 1. Sp Sq = Sq Sp holds for every p, q ∈ π(G), 2. G = ⟨Sp | p ∈ π(G)⟩. For a set σ of primes, recall that σ 󸀠 denotes the complement π(G)\σ, that is, σ∪σ 󸀠 = π(G) and σ ∩ σ 󸀠 = 0. Proposition 2.61. Let G be a prosolvable profinite group and H a closed subgroup. The following statements hold: (a) G has a Sylow base, (b) any two Sylow bases are conjugate in G; in particular, any two σ-Sylow subgroups are conjugate, (c) every Sylow base of H extends to a Sylow base of G, (d) let {Sp : p ∈ π(G)} be a Sylow base of G; if H is normal in G, then (H ∩ Sp )p is a Sylow base of H. Moreover, for any set of primes σ and σ-Sylow subgroup Sσ the intersection H ∩ Sσ is a σ-Sylow subgroup of H. Proof. Statements (a) and (b) are contained in [93, Proposition 2.3.9]. Although a proof of (c) and (d) can be found in [5] we include one for the convenience of the reader. (c) Set π := π(H) and p󸀠 = π \ {p}. Let {Sp : p ∈ π} be the given Sylow base of H. For every p ∈ π fix a p󸀠 -Sylow subgroup Sp󸀠 of H. Then [93, Proposition 2.3.9 (a)] implies that the subgroups Sp∗ := ⋂q∈p󸀠 Sq󸀠 , p ∈ π form a Sylow base of H. Since two Sylow

bases of H are conjugate by (b), there is an h ∈ H such that Sp = (Sp∗ )h = ⋂q∈p󸀠 Sqh󸀠 . It

is no loss of generality to rename the Sqh󸀠 into Sq󸀠 . Then the given Sylow base of H is represented as Sp = ⋂ Sq󸀠 , q∈p󸀠

p ∈ π.

(2.2)

Let us, in order to avoid confusion, denote for a prime q ∈ π(G) the complement π(G) \ {q} by q.̄ For every p ∈ π(H) Corollary 2.3.7 (b) in [93] implies the existence of ̄ a p-Sylow subgroup Tp̄ of G containing Sp󸀠 . For every q ∈ π(G) \ π the same corollary ̄ ensures the existence of a q-Sylow subgroup Tq̄ of G containing H. Defining, for every q ∈ π(G), Tq := ⋂r=q̸ Tr ̄ [93, Proposition 2.3.9 (a)] implies that {Tq : q ∈ π(G)} is a Sylow base for G.

2.6 Sylow bases in inductively prosolvable groups | 45

For p ∈ π we observe that Tp = ⋂ Tq̄ = ( ⋂ Tq̄ ) ∩ ( ⋂ Tq̄ ) ⊇ ( ⋂ Sq󸀠 ) ∩ H = Sp q=p ̸

q∈π\{p}

q∈σ ̸

q∈σ\{p}

in view of equation (2.2). Thus {Tp : p ∈ π(G)} extends the given Sylow base of H. (d) Let {Sp : p ∈ π(G)} be a given Sylow base of G. By (a) there exists a Sylow base, say {Np : p ∈ π(N)}, of N. By (c) one can extend it to a Sylow base, say {Sp∗ : p ∈ π(G)}, of G. By (b) there exists g ∈ G such that for all p ∈ π(G) one has Sp = gSp∗ g −1 . Since conjugation with g induces an automorphism on N deduce that {gNp g −1 : p ∈ π(N)} is a Sylow base of N. Moreover, by construction, gNp g −1 = Sp ∩ N holds for all p in π(N). Thus {Sp ∩ N : p ∈ π(N)} is a Sylow base of N. Lemma 2.62. Let G be a periodic compactly ruled group in which every finitely generated closed subgroup is prosolvable. Then every compact subgroup is prosolvable. Proof. It suffices to assume G to be compact, i. e., profinite. Then let N be any open normal subgroup of G. Since G/N is finite and thus finitely generated our assumptions imply G/N to be solvable. Since this holds for any open normal subgroup N of G and as G = lim G/N with N running through all open normal subgroups of G, the ←󳨀󳨀N prosolvability of G is established. We now prove the main theorem of this section. Theorem 2.63. Let G be a sigma-compact inductively prosolvable compactly ruled group. Then G possesses a Sylow base {Sp : p ∈ π(G)} such that for any σ ⊆ π(G) (a) Sσ := ⟨Sp | p ∈ σ⟩ is a σ-Sylow subgroup of G for any σ ⊆ π(G), (b) Sσ contains a conjugate of every compact σ-Sylow subgroup of G. Proof. Since G is sigma-compact, in view of Proposition 1.3 and of Definition 1.4, there is an ascending countable chain (Gn )n of profinite open subgroups whose union is G. Lemma 2.62 implies that every Gn is prosolvable. Making use of Proposition 2.61 (c), starting from a Sylow base {S1,p : p ∈ π(G1 )}, one can construct inductively a sequence {Sn,p : p ∈ π(Gn )} of respective Sylow bases of Gn such that Gn ∩ Sp,k = Sp,n holds for every prime p ∈ π(Gn ) and all k ≥ n. For every p ∈ π(G) put Sp := ⋃∞ n=1 Sn,p and note that Sp ∩ Gn = Sn,p is closed by construction and since all Gn are open closed, Sp is a closed p-subgroup of G. Let us first check that Sp is a p-Sylow subgroup of G. If this were not the case, there would be a p-Sylow subgroup Tp of G containing Sp properly. Therefore there exists n such that Gn ∩ Tp properly contains Gn ∩ Sp . This, however, is impossible, as by construction Sn ∩ Sp = Sn,p is a p-Sylow subgroup of Gn and thus would have to coincide with Gn ∩ Tp . The next task, in order to verify Definition 2.60, is to prove that Sp Sq = Sq Sp holds for all p and q in π(G). Observing that Sn,p Sk,q ⊆ Sm,p Sm,q = Sm,q Sm,p holds for m :=

46 | 2 Periodic locally compact groups and their Sylow theory max{n, k} one derives Sp Sq = (⋃ Sp,n )(⋃ Sq,k ) n

k

⊆ ⋃ Sm,p Sm,q = ⋃ Sm,q Sm,p m

m

⊆ ⋃ Sk,q Sn,p = Sq Sp . k,n

With exchanged roles of p and q the inclusion Sq Sp ⊆ Sp Sq follows. Hence Sp Sq = Sq Sp follows, as required. Let us now prove (a), from which the fulfillment of all requirements of Definition 2.60 will follow by choosing σ := π(G). Fix σ ⊆ π(G). Then, as {Sn,p : p ∈ π(Gn )} is a Sylow base of Gn it follows that Sn,σ := ⟨Sn,p | p ∈ σ⟩ is a σ-Sylow subgroup of Gn . Observe that Sn,p ⊆ Sn+1,p implies Sn,σ ⊆ Sn+1,σ for all n ∈ ℕ. Therefore Sσ := ⋃∞ n=1 Sn,σ satisfies Sσ ∩ Gn = Sn,σ for every n. Thus Sσ is a closed σ-subgroup of G. We still have to prove that Sσ is a maximal σ-subgroup of G. If this were not the case, then, taking Lemma 2.4 into account, there is a σ-Sylow subgroup Tσ of G containing Sσ properly. Then there is n ∈ ℕ such that Sσ ∩ N = Sn,σ is properly contained in the σ-subgroup Tσ ∩ Gn of Gn . This would contradict the fact that Sn,σ is a σ-Sylow subgroup of Gn . Thus (a) is established. For proving (b), fix a compact σ-subgroup L of G. Then there exists n such that L ⊆ Gn . Since Gn is prosolvable there exists g ∈ G with Lg ⊆ Sn,σ , where Sn,σ is as defined during the proof of (a). As Sn,σ is contained Sσ , (b) is established. Example 2.23 shows that the condition of G being sigma-compact cannot be dropped. In that example (a) is well satisfied without (b) being true. Lemma 2.64. Let G be a periodic inductively prosolvable compactly ruled sigmacompact group and {Sp : p ∈ π(G)} a Sylow base of G. Suppose that N is a closed subgroup of G such that every p-Sylow subgroup of N is normal in G. Then {Sp ∩N : p ∈ π(N)} is a Sylow base for N. Proof. Let S be a p-Sylow subgroup of N. Since S is normal in G, Proposition 2.12 (ii) implies S ⊆ Sp . From the normality of the p-Sylow subgroups of N it follows that N satisfies the assumptions of Theorem 2.53. In particular S = Np and Theorem 2.53 shows that N = ⟨Np | p ∈ π(N)⟩. Thus {Np : p ∈ π(N)} is a Sylow base of N.

3 Abelian periodic groups In this chapter we shall exclusively treat the class of topological abelian groups that have been called periodic. Notation 3.1. Every natural number n has, for a given prime p, a presentation n = pν n󸀠 , where p does not divide n󸀠 . We let νp (n) denote this number ν.

3.1 Braconnier’s theorem Let us recall from Definition 1.13 that a locally compact abelian group G is periodic provided it is totally disconnected and is a union of compact subgroups. Since a totally disconnected locally compact abelian group has arbitrarily small compact open subgroups, in such a group, compact open subgroups abound. Indeed, if C and K are compact open subgroups of a locally compact abelian group, then so are C + K and C ∩ K, and (C + K)/K ≅ C/(C ∩ K) is finite as are all other factor groups in the following diagram. In this sense, all compact open subgroups of a locally compact totally disconnected group are close to each other; we keep this in mind, see Figure 3.1. There is a terminology that defines this sort of closeness of two subgroups. Definition 3.2. Two subgroups C and K of a group G are called commensurable if the index of C ∩ K is finite both in C and in K. Commensurability is easily seen to be an equivalence relation on the set of subgroups of a group. If G is in fact a topological abelian group, then two compact open subgroups are always commensurable. As we consider periodic locally compact abelian groups we emphasize that we have the following basic facts on the set 𝒩 (G) of the compact open subgroups C of G. C+K C

K C∩K

Figure 3.1: Commensurable compact subgroups. https://doi.org/10.1515/9783110599190-003

48 | 3 Abelian periodic groups Fact. A periodic locally compact abelian group G is the directed union of the set 𝒩 (G) and each two of its members are commensurable. Any factor group G/N, N ∈ 𝒩 (G), is discrete, and G is the projective limit limN∈𝒩 (G) G/N. For all of these elementary reasons, the compact open subgroups play a significant role in any general theory of the class of locally compact abelian periodic groups. (The fact that a topological group G which is the union of its compact open subgroups is indeed a directed union, by the way, continues to apply to locally compact solvable groups; see Corollary 1.12 in conjunction with Proposition 1.3.) In order to understand where periodic locally compact abelian groups are positioned in the universe of all locally compact abelian groups the reader should consult Theorems 7.56 and 7.67 in [54], p. 354, respectively, p. 365. Indeed, for a topological group G, denote by comp(G) the set of elements contained in some compact subgroup and with comp0 (G) its connected component of 0. Then for a locally compact abelian group G, comp0 (G) is the unique largest compact connected subgroup and both comp(G) and comp0 (G) are fully characteristic subgroups of G. Now G has what we comp(G) may call the periodic component comp . 0 (G) A subgroup M of a locally compact abelian group G is monothetic if it is algebraically and topologically generated by one element, that is, there is an x such that M = ⟨x⟩. Equivalently, there is an epimorphism (in the category of locally compact abelian groups) e: ℤ → M, and by duality this means that there is a monomorphism ̂ → ℝ/ℤ. Either M ≅ ℤ or else M is compact (see, e. g., [54], Proposition 7.43); M ê: M ̂ is discrete, and thus M is compact monothetic iff M ̂ is isomorphic to a is compact iff M subgroup of the circle group with its discrete topology, usually written (ℝ/ℤ)d . Since a compact abelian group is totally disconnected iff its character group is a torsion group ̂ is isomor(see [54], Corollary 8.5), M is compact totally disconnected monothetic iff M phic to a subgroup of ℚ/ℤ. Now consider a prime number p. An element x in a locally compact abelian group G is a p-element provided the sequence pn x, n ∈ ℕ, tends to the zero element 0 in A. This just re-phrases Definition 2.2. Equivalently, x is a p-element if and only if the monothetic group ⟨x⟩ is compact and is isomorphic either to a finite cyclic p-group or to the group ℤp of p-adic integers, that is, to a quotient group of ℤp (see [2, Lemma 2.11]). Dually, this means that x is a p-element if and only if the character group of ⟨x⟩ is a subgroup of the Prüfer group ℤ(p∞ ) = p1∞ ⋅ℤ/ℤ ⊆ ℚ/ℤ. Note that a p-element x has infinite order if and only if ⟨x⟩ ≅ ℤp . Now we formulate our basic definition (cf. [2, 27]). Recall from Definition 2.2 that a locally compact abelian group G is a p-group provided every element in G is a p-element. Let us note right away that the group ℚp of p-adic rationals is a noncompact locally compact abelian torsion-free p-group which is isomorphic to its dual. From the viewpoint of the theory of locally compact abelian groups, ℚp provides a sharp contrast to the group ℝ of reals as we shall see several times in this text.

3.1 Braconnier’s theorem

| 49

̂ is a A compact abelian group C is a p-group iff its discrete character group C ̂ is a torsion group. Then C is totally disconnected (see, e. g., p-group. In particular, C [54], Corollary 8.5). Therefore every locally compact abelian p-group is periodic. Modulo any of its open compact subgroups C any locally compact abelian p-group G is a discrete p-group G/C in the traditional sense. For a periodic locally compact abelian group G let Gp denote the union of all p-subgroups. Then Gp is a closed and fully characteristic subgroup called the p-primary component; this is indeed the p-Sylow subgroup. But we recognize here that different subspecialties tend to have a different terminology. In order to see the significant relationship of a periodic abelian group G with its Sylow subgroups Gp , recall the concept of a local product from Definiton 2.49. Let us re-phrase a well-known result due to J. Braconnier in [12] to which we shall refer frequently in this text. Theorem 3.3 (J. Braconnier). Let G be a periodic locally compact abelian group and C any compact open subgroup of G. Then G is isomorphic to the local product loc

∏(Gp , Cp ). p

(B)

̂ and since G is periProof. Since G is topologically isomorphic to the dual group of G ̂ ̂ are odic, [47, (24.18) Corollary] implies that G is zero-dimensional. Therefore G and G both zero-dimensional. Then [12, Théorème 1, page 71] immediately yields the desired result. Theorem 3.3 implies that there is an exact sequence of locally compact abelian groups, i. e., 1 → C → G → D → 1, where C = ∏p Cp is profinite and D = ⨁p Dp is a discrete torsion group with Dp = Gp /Cp . The p-primary subgroup of G gives rise to an exact sequence, i. e., 1 → Cp → Gp → Dp → 1. Braconnier’s theorem reduces the structure theory of periodic locally compact abelian groups to the structure theory of locally compact abelian p-groups. Even though the structure of locally compact abelian groups is generally considered to be rather thoroughly known through a wealth of publications, one keeps stepping into corners that are not elucidated in up-to-date literature. In our investigations in [46] we encounter issues about noncompact locally compact abelian groups which do not appear to be discussed in the literature even though some of them were anticipated in Braconnier’s article on his local product [12]. Here we shall treat some of them.

50 | 3 Abelian periodic groups Firstly, we shall deal with some points concerning noncompact locally compact abelian torsion groups. Secondly, we shall discuss some aspects of totally disconnected torsion-free locally compact abelian groups, which one might consider unexpected if not pathological for anyone who should ever expect that the world dominated by ℚp should in any way emulate a world dominated by ℝ. For instance we shall exhibit comparatively tame examples of locally compact abelian p-groups having closed subgroups isomorphic to ℚp which are not direct summand in the sense of topological groups. One guiding idea in dealing with locally compact abelian p-groups is the formation of the divisible hull of a compact open subgroup. For compact abelian groups we often refer to the monograph [54]. It will be convenient to use additive notation for abelian groups. We shall proceed by adhering to the very crude classification into torsion groups, torsion-free groups, and divisible groups among the locally compact abelian p-groups. Prototypes of torsion groups are the compact groups ℤ(pn )I for some set I, the discrete groups ℤ(p∞ )(I) , and the obvious combinations (and some not so obvious combinations) of these; prototypes of torsion-free p-groups are the compact groups ℤIp and the locally compact abelian groups ℚnp for some natural number n, and the divisible ones among these should be obvious. Our formulation of the full classification of locally compact abelian torsion-free p-groups in Theorem 3.51 is far less obvious in the end. Since ℤp is a compact open subgroup of ℚp , torsion-free locally compact abelian p-groups appear to be close to divisible ones, and we shall verify this conjecture. Every torsion-free divisible locally compact abelian p-group is a ℚp -vector space, but it is a topological ℚp -vector space if and only if it is finitedimensional as a ℚp -vector space. We shall prove the nontrivial fact that every torsion-free locally compact abelian group G has a unique largest divisible subgroup div(G) that is closed and decomposes G into a topologically and algebraically direct sum R⊕div(G) with a reduced summand, that is, a group having no nontrivial divisible subgroup. Divisible locally compact abelian torsion groups appear to be a bit more complicated although they can be classified in a sense. In the classes of either discrete or compact p-groups one has the comfortable concept of p-rank. This concept begins to show its difficulties in the example of ℚp which should have p-rank 1. We propose a concept of rank that applies to the full class of locally compact abelian p-groups. We recover Čarin’s classification of finite p-rank p-groups. On the way, its significance is illustrated by its occurrence in exemplary situations.

3.2 Preliminaries about the p-rank In the abstract theory of abelian p-groups one has a clear-cut theory of a rank which is easily transported to the class of compact p-groups via duality. While a general theory

3.2 Preliminaries about the p-rank | 51

of a p-rank for the full class of locally compact abelian p-groups is a surprisingly delicate issue which we shall discuss in detail at a later point, we must review and clarify the concept of the rank of discrete and compact p-groups at this early point. Definition 3.4. For any prime p, an abelian group A has a p-socle Sp (A) = {a ∈ A : p⋅a = 0} = ker μAp , where μAp : A → A is the morphism defined by μp (a) = p⋅a. For a p-group A, the socle is written more simply as S(A). ̂ G = μA . Then ̂ with μ If G is a compact abelian group, this applies to the dual A := G, p p ̂ Accordingly, using the Annihilator Mechanism G/p⋅G = coker μGp = (ker μAp ) ̂ = S(A). of Duality (cf. [54], Theorem 7.64 (vi)) we observe the following. Remark 3.5. We have S(A)⊥ = p⋅G and (p⋅G)⊥ = S(A). Definition 3.6. For a compact abelian group G and its character group A, the cardinal dimGF(p) Sp (A) = ℵ, where G/p⋅G ≅ GF(p)ℵ , is called both the p-rank of A and the p-rank of G, written rankp A = rankp G. In particular, we record the following. Remark 3.7. If G is a compact p-group, then rankp G = rankp (G/p⋅G) = ℵ, where G/p⋅G is a product of ℵ factors ℤ(p). rank (G)

If G is a compact p-group, then there exists a quotient morphism ℤp p → G. (See [57].) For any natural number n, the group ℤnp is generated by n elements, and, accordingly, any compact abelian p-group G of p-rank n is topologically generated by n elements. A finite abelian p-group has finite p-rank iff it is finitely generated and a discrete abelian p-group has finite p-rank iff it is isomorphic to a finite direct sum of summands ≅ ℤ(pn ), n = 1, 2, . . . , ∞. Remark 3.8. (a) Let C be a compact p-group and B a closed subgroup of finite index. Then C has finite p-rank iff B has finite p-rank. (b) Let A be a discrete p-group and E a finite subgroup. Then A has finite p-rank iff A/E has finite p-rank. Proof. The situations (a) and (b) are dual; it suffices to prove (a). If B is finitely generated, then C is finitely generated since C/B is finite. Conversely, assume that C is finitely generated. Then there is a surjective morphism f : ℤnp → C for a natural number n. Then f −1 (B) is a closed subgroup of ℤnp and therefore is isomorphic to ℤm p for

m ≤ n (since dually a quotient of ℤ(p∞ )(n) is a divisible group of p-rank ≤ m). Therefore B is generated by m elements.

52 | 3 Abelian periodic groups While we shall say more about finite p-rank locally compact abelian p-groups later, we make the following observations right away. A locally compact abelian p-group G is finitely generated iff it is a quotient group ̂ is isomorof a group ℤkp for a nonnegative integer k. Thus G is finitely generated iff G ∞ k ∗ ̂ phic to a subgroup of ℤ(p ) . This means that G ≅ D ⊕ F for the maximal divisible ̂ and some finite group F ∗ . Since D ≅ ℤ(p∞ )m for some m ≤ k, passing subgroup D of G to the dual G we see that G is finitely generated iff there is a nonnegative integer m and a finite abelian group F such that G ≅ ℤm p ⊕ F. By what we saw about the p-rank of compact and discrete abelian groups this holds exactly if G is compact of finite p-rank. For easy reference we summarize (using the notation ℕ0 = {0, 1, 2, . . . }).

Proposition 3.9. For a locally compact abelian p-group G the following conditions are equivalent: (1) G is finitely generated, (2) G is compact and rankp (G) is finite, (3) (∃m ∈ ℕ0 , F ≤ G finite) G ≅ ℤm p ⊕ F. The locally compact p-group ℚp is not finitely generated. Its p-rank will be determined later to be 1. For an abelian compact p-group, our definition of p-rank agrees then with the definition of rank of the compact p-group G as given in [93, page 90] for free pro-p groups.

3.3 Locally compact abelian torsion groups Let us begin by recalling some very basic facts on locally compact abelian torsion groups. Clearly, a locally compact abelian torsion group G is periodic, and so it always has compact open subgroups (with any two of them being commensurable). In particular, any locally compact abelian torsion group has compact open abelian torsion subgroups. Every such group has finite exponent by Corollary 8.9 (iii) of [54]. Then the character group of a compact abelian group of finite exponent is a discrete abelian torsion group of finite exponent and therefore is the direct sum of cyclic groups of bounded order (see [32, 33]). By duality, therefore, we have the following observation (cf. [54], Corollary 8.9 (iii)). Lemma 3.10. A compact abelian torsion group is a direct product of cyclic groups of bounded order. Of course, every locally compact abelian torsion group is periodic and thus has its unique Braconnier–Sylow decomposition (B). For the set π of all primes, by Lem-

3.3 Locally compact abelian torsion groups | 53

ma 3.10 the set ϕ := {p ∈ π : Cp ≠ {1}} is finite and so loc

∏ (Gp , Cp ) = ⨁ Gp p∈π\ϕ

p∈π\ϕ

is a discrete torsion group, so that, algebraically and topologically, G is the direct sum of ∏p∈ϕ Gp and ⊕p∉ϕ Gp . We summarize as follows. Proposition 3.11. (i) For any locally compact abelian torsion group G we find a finite set ϕ ⊆ π of primes such that for all p ∈ π \ ϕ the p-primary subgroup Gp is discrete and G ≅ ∏ Gp ⊕ ⨁ Gp . p∈ϕ

(1)

p∈π\ϕ

(ii) For each p ∈ ϕ the group Gp has a compact open subgroup I

I

Cp ≅ ℤ(p)I1 × ℤ(p2 ) 2 × ⋅ ⋅ ⋅ × ℤ(pn ) n

(2)

for a finite collection of sets Ik , k = 1, . . . , n, and Gp /Cp is discrete. Corollary 3.12. For a locally compact abelian torsion group G the following conditions are equivalent: (1) G is discrete, (2) the only compact open subgroups of G are finite, (3) for each prime p, the endomorphism x 󳨃→ p⋅x of Gp is an open map. Proof. Since every locally compact abelian totally disconnected group has a compact open subgroup, (1) and (2) are clearly equivalent. Trivially, (1) implies (3), and so we have to argue that (3) implies (2). By Proposition 3.11 it suffices to show that for each p ∈ ϕ, the compact group Cp is finite. From Proposition 3.11 (2) we know that Cp has exponent pn for some n and thus is a finite product of powers ℤ(pk )Ik for sets I1 , . . . , In . Statement (3) implies that for each k = 1, . . . , n the power I

ℤ(pk−1 ) k = p⋅(

I

k 1 ℤ/ℤ) pk

is open in ℤ(pk )Ik , and that implies that Ik is finite. This proves that Cp is finite. While we see at once from the example G = ∏k∈ℕ ℤ(pk ) that the torsion subgroup tor(G) of a locally compact abelian group need not be closed, some pieces of information are immediately available from duality under the assumption that tor(G) is closed. We need the following simple observation (compare the far more general Theorem I in [1]).

54 | 3 Abelian periodic groups Lemma 3.13. Let D be a discrete divisible subgroup of a totally disconnected locally compact abelian group. Then D is a direct summand in the category of topological groups. Proof. Let U be a compact open subgroup so small that U ∩ D = {0}. Then the sum U + D is direct, algebraically and topologically. Hence there is a continuous projection p: U +D → D. Since D is injective in the category of abelian groups, there is an algebraic extension to a morphism P: G → D which leaves D elementwise fixed. Thus P is a projection. Since U is open in G and P|U = p|U is continuous, so is P. The lemma follows. Now we have an opportunity for a first application of this lemma in the proof of the following observation on closed torsion subgroups. (Compare with [1].) Proposition 3.14. Let G be a locally compact abelian group with a closed torsion subgroup having a compact quotient G/ tor(G). Then tor(G) splits algebraically and topologically. Proof. Since tor(G) ⊆ comp(G) and G/ tor(G) is compact, we know that G = comp(G). ̂ is totally disconnected (see e. g. [54, Corollary 7.68]), and we shall be able to Then G ̂ By duality tor(G) splits iff the annihilator (tor(G))⊥ splits in G ̂ apply Lemma 3.13 to G. (see, e. g., [54, Theorem 7.64 (v)]). Now the compactness of G/ tor(G) implies that its character group (tor(G))⊥ (see, e. g., [54, Theorem 7.64 (i)]) is discrete. Since the compact group G/ tor(G) is torsion-free, this very character group is divisible (see, e. g., [54], Corollary 8.5). By our Lemma 3.13, its being discrete shows that it is a direct summand algebraically and topologically. This applies, in particular, to locally compact abelian p-groups. Note also that for compact groups G the hypothesis that the quotient group G/ tor(G) be compact is automatic. We observe that a compact torsion-free periodic abelian group with tor(G) closed such as G/ tor(G) has a divisible discrete abelian torsion group as character group. In particular, Gp / tor(Gp ) as the dual of a direct sum of copies of Prüfer groups ℤ(p∞ ) is a product of some family of copies of ℤp . If G is an arbitrary locally compact abelian periodic group, then the periodic quotient G/ tor(G) has compact open subgroups. This implies the following corollary. Corollary 3.15. Let G be a periodic locally compact abelian group G whose torsion subgroup tor(G) is closed. Then there is a compact torsion-free subgroup C in G such that the sum C + tor(G) is direct and open in G. If G is a p-group, then C ≅ ℤIp for some set I.

3.3 Locally compact abelian torsion groups | 55

3.3.1 A nonsplit extension of a reduced locally compact abelian p-group by ℚp There is Prüfer’s well-known example; see [32, p. 150] for a nonsplit extension of ℤ(p) by ℤ(p∞ ). Pontryagin duality allows to construct from this a nonsplit extension of ℤp by ℤ(p), namely, the torsion-free group ℤp with subgroup pℤp . Our next example is related to Prüfers example (see Example 7.42) and is a p-adic variation of [54, Theorem A1.32]. Example 3.16. Let p be a fixed prime number. ℕ (ℕ ) In ℤp 0 ⊆ ℤp 0 we consider the elements en defined by en (m) = 1 for m = n and 0 otherwise. (ℕ ) We topologize ℤp 0 as follows. Set ℤp , if n = 0, Cn := { {0}, otherwise. Then C = ∏n∈ℕ0 Cn is a compact subgroup of ℤp 0 and the local product P = ∏loc n∈ℕ0 (Pn , Cn ), Pn = ℤp according to Definition 2.49 is a well-defined nondiscrete locally comℕ

(N )

pact abelian group, algebraically isomorphic to ℤp 0 with a compact open subgroup U = P0 × {0} × ⋅ ⋅ ⋅ ≅ ℤp . We define W to be the ℤp -submodule of P generated by the elements e0 − pn ⋅en , n ∈ ℕ. Then any element w ∈ W is a finite sum w = ∑Nn=1 zn ⋅(e0 − pn ⋅en ). Now N

N

n=1

n=1

w = ∑ zn ⋅e0 − ∑ zn ⋅en ∈ ℤp ⋅e0 ⊕ ⨁ ℤn ⋅en . n∈ℕ

So W ∩ U = {0}, that is, W is discrete, hence closed. Thus if we set ∇p := P/W,

and gn := en + W ∈ ∇p

for n ∈ ℕ0 ,

then ∇p is a locally compact p-group and ℤp -module generated as a ℤp -module by the elements gn , n ∈ ℕ. The function z 󳨃→ z⋅g0 : ℤp → ℤp ⋅g0 is an isomorphism of abelian p-groups and hence of ℤp -modules. The submodule ∇p󸀠 := ℤp ⋅g0 of ∇p is algebraically isomorphic to ℤp as a ℤp -module. In the sequel for X a subset of any ℤp -module we let ⟨X⟩ℤp denote the ℤp -submodule generated by X, i. e., all finite sums of the form s = ∑j∈J zj ⋅ xj . Let s be any element of ∇p . Then s = ∑nm=1 zm ⋅gm . Then for 1 ≤ m ≤ n we have m p ⋅gm = g0 . Hence pn ⋅s ∈ ℤp ⋅g0 ≅ ℤp . Therefore ∇p is a locally compact abelian p-group. We recall that for an abelian group A the subset Div(A) of all divisible elements is a characteristic subgroup, the Ulm subgroup; in any ℤp -module A, this is the set of all elements a ∈ A for which the equation pn ⋅x = a has a solution for all n ∈ ℕ. We recall in passing that div(A) is the unique largest divisible subgroup of A and that

56 | 3 Abelian periodic groups div(A) ⊆ Div(A). We shall call a sequence of roots bn of an element b0 in a ℤp -module B satisfying pn ⋅bn = b0 consecutive if for all natural numbers n ∈ ℕ we have p⋅bn+1 = bn . We now discuss that in the class of locally compact abelian p-groups the group ∇p illustrates both the complications arising with the concept of divisibility and that the p-adic vector group ℚp behaves starkly different from the real vector group ℝ in significant respects. For instance, ∇p = ⟨g0 , g1 , . . .⟩ℤp . Theorem 3.17. The locally compact abelian group ∇p has the following properties: (i) If B is any ℤp -module with a divisible element b0 , then for every set of roots bn ∈ B, n ∈ ℕ, with pn ⋅bn = b0 , there is a unique morphism of ℤp -modules d: ∇p → B with d(gn ) = bn . If B is a locally compact p-group, then d is a continuous morphism of locally compact p-groups. (ii) There is a quotient morphism of locally compact abelian p-groups δ: ∇p → ℚp

defined by

δ(gn ) = 1/pn .

If in (i) the roots bn are consecutive, then d: ∇ → B factors through δ, i. e., there is a unique q: ℚp → B with d = qδ. (iii) We have ℤp ⋅gn ≅ ℤp for all n and g0 = pn ⋅gn . (iv) We have ker δ = tor(∇p ). That is, ∇p / tor ∇p ≅ ℚp and we have the exact sequence 0 → tor(∇p ) → ∇p → ℚp → 0. (v) We shall use the abbreviation ∞

Σp := ⨁ ℤ(pn ). n=1

Then ∇p /∇p󸀠 ≅ Σp and tor(∇p ) ∩ ∇p󸀠 = {0}. In particular, tor(∇p ) is discrete. Moreover, if G is any subgroup with G ∩ ∇p󸀠 = {0}, then G ⊆ tor(∇p ), and therefore G is discrete. (vi) We have div(∇p ) = {0}, i. e., ∇p is reduced. However, Div(∇p ) = ∇p󸀠 . (vii) The subgroup tor(∇p ) is not a direct summand of ∇p . The exact sequence of item (iv) fails to be split. (viii) The group Σp contains a subgroup K ≅ tor(∇p ) such that (a) Σp /K ≅ ℚp /ℤp ≅ ℤ(p∞ ), (b) the subgroup K of Σp is not a direct summand, (c) K and therefore tor(∇p ) is isomorphic to Σp . In particular, we have a nonsplit exact sequence δ

0 → ⨁ ℤ(pn ) → ∇p 󳨀󳨀󳨀󳨀→ ℚp → 0. n∈ℕ

(ℕ )

Proof. (i) Note that ℤp 0 is the free ℤp -module over {en | n ∈ ℕ0 }. So there is a unique morphism of ℤp -modules f : ℤ(ℕ0 ) → ∇p given by f (en ) = bn . We observe that f (e1 −

3.3 Locally compact abelian torsion groups | 57

pn ⋅en ) = f (e1 ) − pn ⋅f (en ) = b1 − pn ⋅bn = 0. Hence f vanishes on W. Thus f induces a unique morphism d: ∇p = ℤ(ℕ) /W → B with d(x + W) = f (x). In particular, d(gn ) = f (en ) = bn for n ∈ ℕ0 . (ii) Statement (i) applies to B = ℚp with bn = 1/pn yielding a unique morphism with δ(gn ) = 1/pn ,

δ: ∇p → ℚp

(1)

mapping ∇p󸀠 isomorphically onto ℤp . Now assume that the bn are consecutive. This means that (∀n ∈ ℕ0 ) p⋅bn+1 = bn . Let ιn : p1n ⋅ℤp →

1 ⋅ℤp pn+1

⋅⋅⋅

1 ⋅ℤp pn

let fn : p1n ⋅ℤp

and κn : ℤp ⋅bn → ℤp ⋅bn+1 be the inclusion morphisms and

→ ℤp ⋅bn be induced by the generator p1n of the free ℤp -module p1n ⋅ℤp being mapped to bn . Then we have an infinite commutative diagram, i. e., ↑ ↑ fn ↑ ↑ ↓ ℤp ⋅bn

⋅⋅⋅ ⋅⋅⋅

ιn

󳨀󳨀󳨀󳨀→ κn

󳨀󳨀󳨀󳨀→

1 ⋅ℤp pn+1

↑ ↑ fn+1 ↑ ↑ ↓ ℤp ⋅bn+1

⋅⋅⋅

ℚp

⋅⋅⋅

⟨b0 , b1 , . . .⟩ℤp ⊆ B,

⋅⋅⋅

in which the horizontal sequences are ascending with their union being at the right ends. The vertical maps induce a unique morphism ℚp → ⟨b0 , b1 , . . . ,⟩ℤp making the diagram commutative and whose coextension is the required morphism q: ℚp → B. (iii) Since δ(gn ) = p1n ∈ ℚ and this element generates an infinite torsion-free group, the compact procyclic p-group ℤp ⋅gn is torsion-free, whence x 󳨃→ x⋅gn : ℤp → ℤp ⋅gn between monothetic p-groups is necessarily an isomorphism. (iv) Let a ∈ ker δ ⊆ ∇p = ⟨g0 , g1 , . . .⟩ℤp , that is, δ(a)=0, where a = ∑n∈ℕ0 zn ⋅gn is a finite sum. So from (1) we have 0 = δ(a) = ∑n∈ℕ0 zn /pn . If N = max{n | zn ≠ 0}, then ∑Nn=0 (pN /pn )zn = 0, or, equivalently,

N

pN z . n n n=1 p

pN z0 = − ∑ (ℕ0 )

Consequently, in ℤp

from (2) we obtain N

N

pN ⋅ ∑ zn ⋅en = pN z0 ⋅e0 + ∑ pN zn ⋅en n=0

n=1

N

N

pN ⋅e n 0 n=1 p

= ∑ pN zn ⋅en − ∑ n=1

N

pN z ⋅(e0 − pn ⋅en ) ∈ W. n n p n=1

=−∑

(2)

58 | 3 Abelian periodic groups Thus a ∈ tor(∇p ), that is, ker δ ⊆ tor(∇p ). However, since ℚp is torsion-free, ker δ ⊇ tor(∇p ). Thus ker δ = tor(∇p ). (v) Since ∇p󸀠 is torsion-free by (iii), we have ∇p󸀠 ∩ tor(∇p ) = {0} and because ∇p󸀠 is open in ∇p , we know tor(∇p ) to be discrete. Also, ∇p

∇p󸀠

≅

⨁∞ n=0

ℤp ⋅en ∞ n ⨁n=0 p ℤp ⋅en

∞

≅ ⨁ ℤ(pn ) = Σp . n=1

It follows that tor(∇p ) is isomorphic to a subgroup of Σp . If a = ∑n∈ℕ0 zn ⋅gn is any element of ∇p with N = max{n | zn ≠ 0}, then pN ⋅gn = pN pn

pn ⋅gn =

pN pn

⋅g0 , whence pN ⋅a ∈ ∇p󸀠 . Thus if G is any subgroup of ∇p not contained in

tor(∇p ), then G must contain an element a of infinite order and we have 0 ≠ pN ⋅a ∈ G ∩ ∇p󸀠 . Therefore, G ⊈ tor(∇p ) implies G ∩ ∇p󸀠 ≠ {0}. n (vi) Claim: The group Σp = ⨁∞ n=1 ℤ(p ) contains no divisible elements. Hence it is reduced. Indeed suppose that a ∈ Σp were a nonzero divisible element, let Σp → ℤ(pn ) be a projection which does not annihilate a. Homomorphisms preserve divisibility. Thus ℤ(pn ) would contain a divisible element b ≠ 0. Then the equation pn ⋅x = b would be solvable in ℤ(pn ). Since pn ⋅x = 0, a contradiction follows. This proves the claim. Hence the image of Div(∇p ) is trivial in ∇p /∇p󸀠 . Thus Div(∇p ) ⊆ ∇p󸀠 . But ∇p󸀠 ≅ ℤp is reduced, and thus div(∇p ) ⊆ Div(∇p ) is zero. Since e1 − pn ⋅en ∈ W we know g0 = pn ⋅gn . Thus ∇p󸀠 ⊆ Div(∇p ). Hence ∇p󸀠 = Div(∇p ). (vii) If tor(∇p ) were a direct summand, then ∇p would contain a subgroup A such that ∇p = tor(∇p ) ⊕ A and A ≅ ∇p / tor(∇p ) ≅ ℚp . But this would contradict the fact that ∇p is reduced. (viii) (a) Set ∇p∗ = ∇p󸀠 + tor(∇p ). Since tor(∇p ) ∩ ∇p󸀠 = {0} by (v), the sum is direct. So ∇p∗ / tor(∇p ) ≅ ∇p󸀠 ≅ ℤp and ∇p∗ /∇p󸀠 ≅ tor(∇p ). By (iv) we have ∇p / tor(∇p ) ≅ ℚp , and for every infinite monothetic subgroup ℤp ⋅q of ℚp , we have ℚp /ℤp ⋅q ≅ ℚp /ℤp . As a consequence, ℚp ℤp

≅

∇p / tor(∇p )

∇p∗ / tor(∇p )

≅

∇p

∇p∗

≅

∇p /∇p󸀠

∇p∗ /∇p󸀠

.

n ∗ 󸀠 By (v) we have an isomorphism ϕ: ∇p /∇p󸀠 → ⨁∞ n=1 ℤ(p ) = Σp . We set K = ϕ(∇p /∇p ) and have assertion (a). (b) In order to prove the second, suppose that K were a direct summand of Σp . Then ∇p∗ /∇p󸀠 = (∇p󸀠 ⊕ tor(∇p ))/∇p󸀠 would be a direct summand of ∇p /∇p󸀠 implying that tor(∇p ) would be a direct summand of ∇p , which is impossible by (vii). (c) This assertion will follow from the following version of Prüfer’s lemma, which will complete the proof.

Lemma 3.18. In Σp := ⨁n∈ℕ ℤ(pn ) assume ℤ(pn ) = ⟨en ⟩ with pn ⋅en = 0. Let K := 󸀠 ∑∞ n=1 ℤp ⋅(en − p⋅en+1 ) and set B = Σp /K. If we define en = en + K in B for n ∈ ℕ, then 󸀠 󸀠 󸀠 󸀠 ∞ p⋅e1 = 0 ≠ e1 and en = p⋅en+1 whence B ≅ ℤ(p ). Then the following statements hold for the quotient morphism f : A → B:

3.3 Locally compact abelian torsion groups | 59

(a) The element en − p⋅en+1 has order pn . (b) The endomorphism ϕ: Σp → Σp defined by ϕ(en ) = en − p⋅en+1 implements an isomorphism onto its image. That is, the sequence ϕ

f

0 → Σp 󳨀󳨀󳨀󳨀→ Σp 󳨀󳨀󳨀󳨀→ B → 0 is exact. Proof. (a) The two elements pn−1 ⋅en and pn ⋅en+1 are two GF(p)-linearly independent elements of the socle of Σp , and so pn−1 ⋅(en − p⋅en+1 ) ≠ 0. But pn ⋅(en − p⋅en+1 ) = 0. (b) The image of ϕ is K by definition. For a proof of the injectivity let a ∈ Σp such that ϕ(a) = 0. Now a = ∑ zn ⋅en with a finite support sequence of elements zn ∈ ℤp . ∞ ∞ ∞ ∞ Then 0 = ∑∞ n=1 zn ⋅en − ∑n=1 pzn ⋅en+1 = ∑n=1 zn ⋅en − ∑n=2 pzn−1 ⋅en = z1 .e1 + ∑n=2 (zn − pzn−1 )⋅en . In the direct sum Σp of the ℤp -submodules ⟨en ⟩ℤp this implies z1 = 0, z2 − pz1 = 0, z3 − pz2 = 0, and so on. Inductively, this shows 0 = z1 = z2 = z3 = ⋅ ⋅ ⋅ , and so a = 0. This completes the proof of (b). We can iterate ϕ and set Sn = ϕn (Σp ), n = 0, 1, 2, . . . . Then Σp = S0 ⊇ S1 ⊇ S2 ⊇ ⋅ ⋅ ⋅ . Since ϕ is injective, all Sn are isomorphic to Σp . Proposition 3.19. The countable torsion group Σp is filtered by a sequence S0 = Σp ⊇ S1 ⊇ S2 ⊇ ⋅ ⋅ ⋅ of isomorphic subgroups such that (i) Sn−1 /Sn ≅ ℤ(p∞ ) for n ∈ ℕ, (ii) ⋂n∈ℕ Sn = {0}. Proof. We have to prove (i) and (ii). For each n ∈ ℕ, set Kn = Sn−1 /Sn ; in particular K0 = ℤ(p∞ ). The injective endomorphism ϕ: S0 → S0 leaves Sn invariant and induces an injective endomorphism ϕn : Sn → Sn with cokernel Kn . We have the commutative diagram 0 ↓ 0 ↓ .. . 0 .. .

→ →

→

S0 ↓ S1 ↓ .. . Sn .. .

ϕ

󳨀󳨀󳨀󳨀→ ϕ2

󳨀󳨀󳨀󳨀→

ϕn

󳨀󳨀󳨀󳨀→

S0 ↓ S1 ↓ .. . Sn .. .

→ →

→

K0 ↓ K1 ↓ .. . Kn .. .

→ →

→

0 ↓ 0 ↓ .. . 0 .. .

in which all rows are exact and the vertical morphisms Sn−1 → Sn n ∈ ℕ are the isomorphisms induced by ϕ|Sn . Since the downward facing arrows Sn−1 → Sn are isomorphisms inductively and K1 = ℤ(p∞ ), it follows, inductively, that Kn ≅ ℤ(p∞ ) for all n ∈ ℕ.

60 | 3 Abelian periodic groups (ii) By the definition of ϕ in Lemma 3.18 we have ϕ(en ) = en − p⋅en+1 . We define ℓ: Σp → ℕ as follows: Let x = ∑n∈ℕ xm with xm ∈ ℤ(pm ). Then 0, if x = 0, { { { ℓ(x) = {max{m ∈ ℕ| 0 ≠ xm ∈ ℤ(pm )} { { m { − min{m ∈ ℕ| 0 ≠ xm ∈ ℤ(p )} + 1, otherwise. In the definition of ϕ in Lemma 3.18 we set ϕ(en ) = en − p⋅en+1 . Thus let y = ∑n∈ℕ yn be ϕ(x) and assume x ≠ 0. Then min{m ∈ ℕ| 0 ≠ ym ∈ ℤ(pm )} = min{m ∈ ℕ| 0 ≠ xm ∈ ℤ(pm )} and max{m ∈ ℕ| 0 ≠ ym ∈ ℤ(pm )} = max{m ∈ ℕ| 0 ≠ xm ∈ ℤ(pm )} + 1. Thus ℓ(ϕn (x)) = ℓ(x) + n.

(3.1)

Now assume that y ∈ ⋂m∈ℕ Sm . Suppose that y ≠ 0 and set n = ℓ(y) ∈ ℕ. Then y ∈ ⋂m∈ℕ Sm ⊆ Sn , and so there is an x ≠ 0 such that ϕn (x) = y. Thus (3.1) shows that ℓ(y) = ℓ(ϕn (x)) = ℓ(x) + n = ℓ(x) + ℓ(y), that is, ℓ(x) = 0 and hence x = 0, which is impossible. This proposition dualizes comfortably according to the annihilator mechanism of locally compact abelian groups (see [54], Lemma 7.17 ff., notably Corollary 7.22, all of which fully applies to locally compact abelian groups). So let P = ℤℕ p be the dual of ⊥ Σp and let Hn ≤ P be the annihilator (Sn ) of Sn ≤ Σp . Since the Sn are descending, the Hn are ascending, and since ⋂n∈ℕ Sn = {0} we know that P = ⋃ Hn . n∈ℕ

(3.2)

For all n ∈ ℕ we deduce via duality from Sn−1 /Sn ≅ ℤ(p∞ ) that Hn /Hn−1 ≅ ℤp for n ∈ ℕ. However, at this point we can utilize the fact that in the category of compact p-groups, the group ℤp is projective (since its dual ℤ(p∞ ) is divisible and hence injective in the category of discrete p-groups; see also [54, Theorem 8.78]). Therefore, for each n ∈ ℕ, the compact group Hn contains a compact subgroup Kn ≅ ℤp such that (∀n ∈ ℕ) Hn = Hn−1 Kn ≅ Hn−1 × Kn .

(3.3)

By induction we conclude at once that (∀n ∈ ℕ) Hn = C1 ⋅ ⋅ ⋅ Cn = C1 × ⋅ ⋅ ⋅ × Cn ≅ ℤnp

(3.4)

3.3 Locally compact abelian torsion groups | 61

and there are algebraic isomorphisms ⋃ Hn = ⟨ ⋃ Cn ⟩ ≅ ⨁ Cn ≅ ℤ(ℕ) p .

n∈ℕ

n∈ℕ

n∈ℕ

(3.5)

Let us collect this information. Corollary 3.20. The group P = ℤ(p)×ℤ(p2 )×ℤ(p3 )×⋅ ⋅ ⋅ contains a dense ℤp -submodule which is algebraically isomorphic to the ℤp -module ℤ(ℕ) p . Corollary 3.21. For any family (nj )j∈J of natural numbers, the profinite p-group G = ∏j∈J ℤ(pnj ) is either a torsion group or else it contains a ℤp -submodule isomorphic to 2 3 ℤ(ℕ) p whose closure is isomorphic to P = ℤ(p) × ℤ(p ) × ℤ(p ) × ⋅ ⋅ ⋅ . Proof. Either the family (nj )j∈J is bounded, in which case G is a torsion group, or else it is unbounded. In that case there is an increasing unbounded subsequence (nj(m) )m∈ℕ . Set km = nj(m) . Since the kn are increasing, we have n ≤ kn . The cyclic group ℤ(pkm ) = ℤ(pnj(m) ) contains a subgroup Bm ≅ ℤ(pm ). Then the group B1 × B2 × B3 × ⋅ ⋅ ⋅ is clearly isomorphic to a subgroup B of G which is isomorphic to ℤ(p) × ℤ(p2 ) × ℤ(p3 ) × ⋅ ⋅ ⋅. Then it follows from Corollary 3.20 that B contains a dense ℤp -submodule algebraically isomorphic to ℤ(ℕ) p , as asserted. Concerning non-splitting ℚp one obtains by using duality the following corollary to Theorem 3.17. ̂p has a closed subgroup Q ≅ ℚp which Corollary 3.22. The locally compact p-group ∇ ̂ does not split but is such that ∇p /Q ≅ ∏n∈ℕ ℤ(p). That is, there is a nonsplit exact sequence ̂p → ∏ ℤ(pn ) → 0. 0 → ℚp → ∇ n∈ℕ

It is noteworthy that Example 3.16 has the following consequence in view of Proposition 3.14; the notation is that of Theorem 3.17. Corollary 3.23. For each natural nonnegative integer n ∈ ℕ0 let Un = δ−1 (

1 ⋅ℤ ) ⊆ ∇p . pn p

Then (1) Each of the subgroups U1 ⊂ U2 ⊂ ⋅ ⋅ ⋅ is open and ∇p = ⋃n∈ℕ0 Un . (2) For each n ∈ ℕ0 there is a compact open subgroup Cn ⊆ ∇p with Cn ≅ ℤp such that Un = Cn ⊕ tor(∇p ). It is a contrast to Proposition 3.14; Corollary 3.15 applies to it for Corollary 3.23.

62 | 3 Abelian periodic groups It is noteworthy that in the Chabauty space 𝒮𝒰ℬ(∇p ) of ∇p we have limn→∞ Un = ∇p due to the very Definition 1.16. The sequence (Cn )n∈ℕ0 has a convergent subnet by Lemma 1.17. Its limit C meets tor(∇p ) trivially, but C ⊕ tor(∇p ) cannot be ∇p since tor(∇p ) does not split in ∇p . Example 3.24. Let Σp := ⨁j≥1 ℤ(pj ) be discrete and fix a generator 1k for ℤ(pj ). Consider the cartesian product of locally compact abelian p-groups U := ℚp × Σp and inside the closed subgroup G topologically generated by the elements hj := (

1 , 1j ), pj

j ≥ 0.

For the sake of convenience set 10 := 0. Since, in G, for all j ≥ 1 the relation pj (

1 , 1j ) = (1, 0) pj

holds, the premises of Theorem 3.17 hold for bj := (

1 , 1j ). pj

Therefore the lemma implies the existence of a ℤp -module homomorphism d : ∇p → G sending, for j ∈ ℕ0 , the element gj ∈ ∇p to bj . Lemma 3.25. The ℤp -homomorphism d : ∇p → G is an isomorphism of locally compact abelian p-groups. Moreover, G is an open subgroup of U = ℚp × Σp . Proof. Every element z in the ℤp -module ∇p has a presentation N

z = ∑ zj gj , j=0

for some N ≥ 0 and coefficients zj ∈ ℤp . If z ∈ ker(d), then N

(0, 0) = d(z) = ∑ zj ( j=0

N z N 1 j , 1 ) = ( , ∑ zj 1j ) ∑ j j pj p j=0 j=0

and the fact that 1j has order pj imply that pj divides zj and therefore zj = pj zj󸀠 for some zj󸀠 ∈ ℤp . The first coordinate yields N−1

zN󸀠 = − ∑ zj󸀠 = 0. j=0

3.3 Locally compact abelian torsion groups | 63

If N = 0 deduce from this that z = 0. Otherwise taking this equation and the relations pj gj = g0 for j ≥ 1 into account one finds N−1

N−1

j=0

j=0

z = ∑ zj󸀠 pj gj − ( ∑ zj󸀠 )pN gN = 0. Finally note that the restriction of d to the open compact subgroup ∇p󸀠 = ⟨g0 ⟩ of ∇p renders a topological isomorphism onto the open compact subgroup ⟨(1, 0)⟩ ≅ ℤp of G. Hence d is a topological isomorphism from ∇p onto G. For proving the additional statement it suffices to remark that ⟨(1, 0)⟩ is actually an open compact subgroup of U and contained in G. Hence G itself is open in U. Many of the preceding results were derived from our knowledge of compact torsion p-groups, notably the fact that a compact abelian torsion group has a finite exponent. The details of the structure of noncompact and nondiscrete locally compact abelian torsion p-groups remains a challenge. We shall see in the next section that every locally compact abelian periodic abelian group G has a divisible hull DG and locally compact abelian divisible torsion p-groups we shall describe completely and explicitly. Proposition 3.11 immediately applies to any locally compact abelian p-group of exponent p and yields the following. Corollary 3.26. Any exponent p locally compact abelian group G is a direct product of a compact and a discrete exponent p-subgroup. Specifically, there are sets I1 and I2 such that G ≅ ℤ(p)(I1 ) ⊕ ℤ(p)I2 .

(R)

The equation (R) anticipates a more generally applicable definition of the p-rank than the one we had in Definition 3.6. That definition yields the two pieces of information rankp (ℤ(p)(I1 ) ) = card(I1 ) and rankp (ℤ(p)I2 ) = card(I2 ). Our later definition of the general p-rank (see Section 3.10) will yield rankp G = card I1 + card I2 . Definition 3.27. A locally compact abelian group G is densely divisible if it contains a dense divisible subgroup. This terminology has been introduced in [3] and here is an example of such a group. Example 3.28. Let ℤ(p) :=

1 ℤ/ℤ p

⊆

1 ℤ/ℤ p∞

= ℤ(p∞ ) ≅

ℤ(p))loc,ℕ . Then (a) G is a locally compact abelian torsion p-group, (b) G has a compact open socle S(G) = ℤ(p)ℕ of rank ℵ0 ,

ℚp . ℤp

Consider G = (ℤ(p∞ ),

64 | 3 Abelian periodic groups (c) the discrete factor group G/S(G) ≅ ℤ(p∞ )(ℕ) is a divisible torsion p-group of rank ℵ0 , (d) the unique largest divisible group D = ℤ(p∞ )(ℕ) of G is countable, dense, and nonclosed, (e) the group D ∩ S(G) = S(D) = ℤ(p)(ℕ) is a countable GF(p)-vector subspace of S(G) and therefore has an algebraic GF(p)-vector space complement C of continuum dimension, that is, algebraically, C ≅ ℤ(p)ℕ , (f) algebraically, S(G)/S(D) ≅ ℤ(p)ℕ . The venue in which Example 3.28 takes place is the abelian divisible torsion group Δ = ℤ(p∞ )ℕ in which we consider the subgroup S := ℤ(p)ℕ , a GF(p)-vector space, whose dimension dimGF(p) S equals card S = 2ℵ0 = the cardinality of the continuum, say, card ℝ. We define on Δ again the finest group topology for which S has its own product topology and is an open subgroup in Δ. Indeed, the group considered in Example 3.28 is an open subgroup of Δ. We consider the Prüfer group ℤ(p∞ ) = (1/p∞ )⋅ℤ/ℤ and the subgroup ℤ(p) = (1/p)⋅ℤ/ℤ. Example 3.29. The group Δ = ℤ(p∞ )ℕ is a divisible locally compact abelian group whose topology is defined by declaring the subgroup S = ℤ(p)ℕ with its compact product topology an open subgroup of Δ. We define inside Δ the subgroup ∞

D := ⋃ ( n=0

ℕ

1 ℕ ℤ/ℤ) ⊆ ℤ(p∞ ) . pn

Then D contains precisely the elements d = (z1 , z2 , . . . ), zk ∈ ℤ(p∞ ), k ∈ ℕ, for which there is an n ∈ ℕ such that pn ⋅d = 0, that is, D is the torsion subgroup of Δ. Moreover, S ⊆ D ⊆ Δ and the following statements hold: (a) D is a nondiscrete locally compact abelian divisible torsion p-group, while D∗ := Δ/D is a discrete torsion-free divisible group, that is, is a ℚ-vector group failing to be a p-group; further, dimℚ D∗ = card D∗ , (b) the socle S of D is a compact open subgroup, (c) the cardinality of D, Δ, and D∗ is the continuum 2ℵ0 , whence D∗ ≅ ℚ(ℝ) , (d) both D/S and Δ/S are discrete divisible abelian groups algebraically isomorphic to D, respectively, Δ, (e) the weights w(D) and w(Δ) (see, e. g., [54], Definition A4.7) agree and equal 2ℵ0 , while (f) both D and Δ are first countable, (g) D is a divisible hull of S, (h) since D is an open divisible subgroup of Δ, we have Δ ≅ D ⊕ D∗ algebraically and topologically. In particular, Δ is not sigma-compact.

3.3 Locally compact abelian torsion groups | 65

It should be clear that nonisomorphic variations of this theme abound. The compact abelian torsion groups showed the significance of a group having finite exponent. In order to apply efficiently the hypothesis of having finite exponent to locally compact abelian periodic groups we introduce an appropriate concept in the following definition, for which we recall that every locally compact abelian periodic abelian group G has a filter basis 𝒩 (G) of compact open subgroups N converging to 0 and satisfying ⋃ 𝒩 (G) = G. Definition 3.30. We say that a locally compact abelian p-group has approximately finite exponents if for each N ∈ 𝒩 (G) there is a natural number n(N) such that pn(N) ⋅G ⊆ N, that is, that G/N has finite exponent. We note the following right away. Remark 3.31. Every profinite p-group has approximately finite exponents. Recall (e. g., from [54], Definition A1.22 ff.) that a subgroup P of an abelian group A is pure if P ∩ n⋅A ⊆ n⋅P for all natural numbers n. In the case of p-groups this is equivalent to P ∩ pm ⋅A ⊆ pm ⋅P for all natural numbers m. The formulation of the following result is guided by a lemma due to Kulikoff (see, e. g., [32], Theorem 27.5). Remark 3.32 (Kulikoff’s lemma). A pure subgroup of finite exponent in an abelian group is a direct summand. Since in a p-group of finite exponent every finite subgroup is contained in a pure subgroup of the same rank, we note the following. Lemma 3.33. A finite subgroup of an abelian p-group of finite exponent is contained in a direct summand of the same p-rank. The compact Hausdorff space 𝒮𝒰ℬ(G) of all closed subgroups of a locally compact group G, that is, the Chabauty-space of G (see Definition 1.16) will play an essential role in the nontrivial proof of the following result. Theorem 3.34. In a locally compact abelian p-group having approximately finite exponents, every finitely generated subgroup is contained in a finitely generated (algebraic and topological) direct summand of the same rank. Proof. Assume that G satisfies the hypotheses of the theorem and let 𝒩 denote the filter basis of all compact open subgroups of G. Thus lim 𝒩 = 0.

(1)

Now let H be a finitely generated subgroup of G of p-rank n. Then, by Proposition 3.9, we have H ≅ ℤnp ⊕ F for some nonnegative integer n and a finite group F. Now H ∗ := ℤnp ⊕ F has arbitrarily small compact open subgroups of the form M =

66 | 3 Abelian periodic groups (pk ℤp )n ⊕ {0}, whence H ∗ /M ≅ ℤ(pk )k ⊕ F, and so rankp (H ∗ /M) = rankp (H ∗ ). Thus for all sufficiently small N ∈ 𝒩 we have rankp (H + N)/N = rankp H/(H ∩ N) = rankp H = n. Thus the set 𝒩H of these N is cofinal in 𝒩 . For each N ∈ 𝒩H , since G has approximately finite exponents, G/N has finite exponent. The quotient (H +N)/N ≅ H/(H ∩N) is finite since H ∩ N is open in H. Moreover, for all N ∈ 𝒩H we have rankp (H/(H ∩ N)) = n.

(2)

Therefore, Lemma 3.33 applies to (H + N)/N ≅ H and G/N and yields subgroups FN and BN of G containing N such that H ⊆ FN

and

rankp (FN /N) = n

(3)

for n = rankp H and G/N = FN /N ⊕ BN /N.

(4)

Now we invoke the compactness of 𝒮𝒰ℬ(G) and find some cofinal function j 󳨃→ Nj : J → 𝒩H for some directed poset J such that the limit (F, B) = limj∈J (FNj , BNj ) exists in 𝒮𝒰ℬ(G) × 𝒮𝒰ℬ(G). We claim G =F⊕B

(4’)

and H⊆F

and

rankp F = rankp H.

(3’)

First we prove (4󸀠 ). From relation (4) we know G = FN +BN , which implies G = F+B. Now let g ∈ F ∩ B. Then g = limi fi = limi bi for some fi ∈ FNj and bi ∈ BNj for a i i suitable cofinal function i 󳨃→ ji : I → J. Now let N ∈ 𝒩H . Then N is a compact open neighborhood of 0. Since limi (bi − fi ) = g − g = 0 there is an i0 ∈ I such that i0 ≤ i implies bi − fi ∈ N. Hence bi = fi + ni with some ni ∈ N ⊆ FN and so bi ∈ FN ∩ BN = N. Since N is compact, g = limi bi ∈ N. Since N was arbitrary in 𝒩H , g = 0 follows. Hence F ∩ B = {0}. Now we prove (3󸀠 ). Since the graph of the containment relation ⊆, defined on 𝒮𝒰ℬ(G), is a closed subset of 𝒮𝒰ℬ(G) × 𝒮𝒰ℬ(G), the containment H + N ⊆ FN for all N implies H ⊆ F. Thus n = rankp H ≤ rankp F and it remains to show the reverse inequality rankp F ≤ n. By (3) there is a n-element subset XN ⊆ FN such that ⟨XN + N/N⟩ = FN /N. We let EN = ⟨XN ⟩. Then FN = EN +N and rankp EN ≤ n. There is a cofinal function k 󳨃→ jk : I → J of directed sets such that the net (ENj )k∈K converges in the compact space 𝒮𝒰ℬ(G) to k a compact p-group E. Since |XNj | = n for all j we have rankp ⟨E⟩ ≤ n.

(i)

3.4 Purity used partially | 67

On the other hand, since limj Nj = {0} we observe E = lim ENj = lim(ENj + Njk ) = lim FNj = F. k

k

k

k

k

k

(ii)

From (i) and (ii) we obtain rankp F ≤ n, which we had to show. As a consequence of Theorem 3.34, we obtain for locally compact abelian p-groups the following variation of the theme of Kulikoff’s lemma; see Remark 3.32. Theorem 3.35. Let G be a locally compact abelian p-group having approximately finite exponents and let A be a finitely generated pure subgroup of G. Then A is a direct summand, algebraically and topologically. Proof. By Theorem 3.34, we have G = F ⊕ B such that A ⊆ F and rankp A = rankp F. As a direct summand, F is pure in B, and since A is assumed to be pure, we conclude A = F. When we now look at the concept of purity of subgroups of p-groups a bit more closely, we shall be able to sharpen this result.

3.4 Purity used partially In a locally compact abelian group G pure subgroups need not be closed (see e. g. [104]). For finite p-rank periodic locally compact abelian groups we have the following observation on partial purity of a subgroup; we shall use it presently. Proposition 3.36. Let G be a locally compact abelian p-group of finite p-rank and A a subgroup satisfying (1) rankp (A) = rankp (G), (2) A ∩ pG ⊆ pA. Then A = G. Proof. Suppose, by contradiction, the existence of x ∈ G \ A. We cannot have ⟨x⟩ ∩ A = {0}, else the p-rank of ⟨A, x⟩ = A ⊕ ⟨x⟩ would exceed the p-rank of G, a contradiction. In particular it follows that the socle of G is a subgroup of A. Then, replacing x by pr x for suitable r ∈ ℕ0 , one can assume that px ∈ A while x ∈ ̸ A. By our assumptions there exists a ∈ A with px = pa. Setting s := a − x one observes that s belongs to the socle of G and hence to A. The contradiction x ∈ A follows. Since we are dealing mostly with locally compact abelian p-groups we earlier reformulated purity for subgroups of these groups explicitly. Definition 3.37. A subgroup A of a locally compact abelian p-group p is pure if (∀m ∈ ℕ) pm ⋅G ∩ A ⊆ pm ⋅A.

(P)

68 | 3 Abelian periodic groups In Proposition 3.36 we got away with the condition p⋅G ∩ A ⊆ p⋅A.

(P1 )

Clearly Condition (P) implies Condition (P1 ). But the converse fails in general as the following example shows. Let G = ℤp ⊕ ℤ(p) and consider the natural epimorphism f : ℤp → ℤ(p) ≅ ℤp /pℤp . Note that ker f = pℤp . Let A denote the closed subgroup A = {(pz, f (z)) : z ∈ ℤp }. Then p⋅G = pℤp × {0} and thus p⋅G ∩ A = {(pz, f (z)) : z ∈ ker f } = p2 ℤp × {0} = p⋅A ⊆ A. However, p2 ⋅G ∩ A = {(pz, f (z)) : pz ∈ p2 ℤp } = p2 ⋅ℤp × {0} and p2 ⋅A = p3 ℤp × {0}, which shows that p2 ⋅G ∩ A ⊈ p2 ⋅A. Thus we keep in mind that Condition (P1 ) is strictly weaker than purity in locally compact abelian p-groups. As a consequence of this theorem and Proposition 3.36, we obtain for locally compact abelian p-groups the following version of Kulikoff’s lemma (see Remark 3.32). Theorem 3.38. Let G be a locally compact abelian p-group having approximately finite exponents and let A be a finitely generated subgroup of G such that A ∩ pG = pA.

(P1 )

Then A is a direct summand. Since every profinite group has approximately finite exponents we have the following conclusion. Corollary 3.39. Let G be an abelian profinite p-group and let A be a finitely generated subgroup of G such that A ∩ pG = pA.

(P1 )

Then A is a direct summand. Corollary 3.40. Let S be a finitely generated subgroup of the locally compact abelian p-group G of approximately finite exponents. Then there is an open subgroup H containing S such that S ∩ pH = pS. Proof. Since G has approximately finite exponents, Theorem 3.34 guarantees the existence of a direct decomposition, i. e., algebraically and topologically G = T ⊕ R, such that T contains S and, by Proposition 3.9, rankp (S) = rankp (T). Hence S is an open subgroup of T and thus H := S ⊕ R serves the purpose.

3.5 Locally compact abelian divisible groups | 69

3.5 Locally compact abelian divisible groups The classification of torsion-free divisible locally compact abelian groups in general is due to G. W. Mackey; see [72] and [47, (25.33)]. A review and reformulation is justified by various clarifications through our descriptions which were motivated, among other things, by our applications in Part III. Via duality, torsion and divisibility are juxtaposed in the context of compact and discrete abelian groups as is illustrated in [54] in the first section of Chapter 8. We pursue this in the context of locally compact abelian p-groups. Braconnier’s decomposition Theorem 3.3 into primary components tells us that the restriction to p-groups is no restriction of generality for periodic abelian groups. We shall have to invoke from the theory of abstract abelian groups the concept of a divisible hull which we recall now (see for instance [54], notably Proposition A1.33 and Corollary A1.36). We shall keep these concepts in mind. Remark 3.41. Any abelian group A is a subgroup of an abelian divisible group D, called a divisible hull of A, such that the following statements hold: (i) every nonzero subgroup of D meets A nontrivially, (ii) card D = max{ℵ0 , card A}, and so (iii) the p-ranks of A and D agree for all primes p, (iv) if D1 and D2 are divisible hulls of A inside an abelian group G, there is an isomorphism f : D1 → D2 such that f |A = idA , (v) if A is torsion-free, then ℚ ⊗ A is a divisible hull of A and thus is unique (up to isomorphism). If A is a subgroup of a divisible group H and D is a divisible hull of A, then the inclusion morphism j: A → H extends to a morphism f : D → H since H is injective as a divisible group, and A ∩ ker f = ker j = {0} implies ker f = {0} by Remark 3.41 (i). Thus H contains an isomorphic copy of D containing A. We now observe that there is a corresponding concept of a divisible hull for locally compact abelian p-groups in the context of topological abelian groups. Proposition 3.42. Let A be a locally compact abelian p-group and D an algebraic divisible hull containing A. We give D the unique group topology for which A is an open subgroup. Then D is a locally compact abelian divisible p-group. Proof. Our definition of the topology on D makes D a locally compact abelian group. In order to show that D is a p-group, we take an arbitrary element x ∈ D and must show that H := ⟨x⟩ is a compact p-group. By Remark 3.41 we find a nonzero element a ∈ A ∩ ⟨x⟩ such that n⋅x = a for some natural number n. By Weil’s lemma (see, e. g., [54], Proposition 7.43), we have the following two cases: Case (1): H ≅ ℤ with the discrete topology. Case (2): H is compact monothetic.

70 | 3 Abelian periodic groups In case (1), n⋅H = ⟨n⋅x⟩ is a nonsingleton discrete infinite cyclic subgroup of A which is impossible since A is a locally compact abelian p-group. Thus D is periodic and thus by Braconnier’s local product Theorem 3.3 is of the form loc

D = ∏ (Dq , Cq ) q∈π

(B)

for the q-Sylow subgroups of D and C is a compact open subgroup of A. We know that A ⊆ Dp since A is a p-group and Dp is the unique largest p-group in D. Assume that Dq ≠ {0} for a prime q. We may consider Dq to be a subgroup of D. Then Dp ∩ Dq ⊇ A ∩ Dq ≠ {0} by Remark 3.41(i). But that implies p = q, and so D = Dp , showing that D is a p-group. In particular H is a compact monothetic p-group. In particular, every locally compact abelian p-group is an open subgroup of a divisible locally compact abelian p-group. Therefore divisible locally compact abelian groups are all but rare. Our Examples 3.28 and 3.29 above were early hints that plausible expectations suggested by the discrete or the compact situations may fail in the nondiscrete and noncompact locally compact one. This cautionary remark also applies to the following example. The verifications of their properties are easy exercises. Let us now look at the following. Example 3.43. Let J be any set and give ℚJp the topology generated by the open sets of the product topology and ℤJp as an additional open subset. Call the resulting topological group G. Then (a) G is a locally compact abelian group with respect to addition, (b) ℤJp is a compact open p-subgroup, and U compact and open in G implies U ≅ ℤJp , (c) G is torsion-free divisible, (d) with respect to componentwise addition, multiplication, and scalar multiplication with p-adic rationals, the abstract group G is in fact a ℚp -algebra, 1 J (e) let D := ⋃∞ n=0 pn ℤp ; then D is an open subgroup of G which is proper if and only if J is infinite; it satisfies conditions (a)–(d) with D in place of G, (f) the subgroup D is the smallest ℚ-vector subspace of G containing ℤJp , namely, the divisible hull ℚ ⊗ ℤJp , (g) D is a locally compact abelian p-group, (h) let μp : G → G again denote the continuous multiplication by p; then the function μp : G → G is an open map if and only if J is finite. This is the case if and only if the function x 󳨃→ p1 ⋅x: G → G is a homeomorphism. This formulation will eventually agree with a general concept of p-rank of locally compact abelian p-groups. From statement (h) we note the following.

3.5 Locally compact abelian divisible groups | 71

Remark 3.44. The torsion-free divisible locally compact abelian p-group D of Statement 3.43 (e) is a topological ℚp -vector space iff it has finite p-rank, that is, iff it is isomorphic to ℚnp for some nonnegative integer n. The aspects of the example discussed in Example 3.43 have a sobering effect. Still we can formulate and prove some general results. Proposition 3.45. Let G be a divisible locally compact abelian p-group and C any compact open subgroup. Let DC be a locally compact abelian divisible hull of C according to Proposition 3.42. Then G is isomorphic to the direct product DC × G/DC for a discrete divisible group G/DC . Proof. A divisible hull DC of a subgroup C of a divisible group G may be considered as a subgroup of G, so without loss of generality we may assume DC ⊆ G. As a divisible subgroup of G, the group DC is algebraically a direct summand of G; however, it contains C and thus is open in G and thus is a direct summand also topologically, that is, G is isomorphic to the direct product DC ×G/DC . As a homomorphic image of a divisible group, G/DC is divisible. We obtained the discrete direct summand ≅ G/DC as a quotient. Let us recall from Lemma 3.13, however, that any discrete divisible subgroup of a totally disconnected locally compact abelian group is always a direct summand algebraically and topologically. Moreover, a discrete divisible p-group is always a direct sum of Prüfer groups; Proposition 3.45 thus has an immediate corollary for p-groups as follows. Theorem 3.46. Let G be a divisible locally compact abelian p-group. Then for each compact open subgroup C, there is a divisible hull DC of C contained in G and a discrete subgroup P ≅ ℤ(p∞ )(J) for some set J such that G = DC ⊕ P. Notice that P is maximal within the set of divisible subgroups meeting C trivially, while G may not have any largest discrete divisible subgroup. Theorem 3.46 tells us that we know divisible locally compact abelian p-groups to the extent that we know divisible hulls of profinite p-groups C. Definition 3.47. We call a profinite abelian group C coreduced if it does not have ℤp as homomorphic image. ̂ is reduced, that is, has no nontrivial divisible subBy duality, C is coreduced iff C ̂ group. Since C is a direct sum of the maximal divisible subgroup and a reduced group, every compact abelian group is a direct product of a compact torsion-free subgroup and a coreduced subgroup. Corollary 3.48. Any divisible locally compact abelian p-group is isomorphic to a direct product of a torsion-free divisible p-group, a divisible hull of a coreduced profinite p-group C, and a discrete divisible p-group.

72 | 3 Abelian periodic groups Proof. By Proposition 3.45 a divisible locally compact abelian group G is isomorphic to the direct product DC ×L for some compact open subgroup C and a discrete divisible group L. So we may assume G = DC . The profinite group C is a product of a compact torsion-free subgroup Cf and a coreduced subgroup Cr . We write C = Cf × Cr . Then DC ≅ DCf × DCr ⊇ Cf × Cr = C. The divisible hull DCf of a torsion-free group is torsion-free, and DCr is a divisible hull of a coreduced profinite group. Since we shall completely classify torsion-free divisible p-groups and since a discrete divisible p-group is of the form ℤ(p∞ )(J) for some set J, the classification problem rests entirely with divisible hulls of coreduced profinite groups. Notably, all divisible torsion groups are of this type. These we shall classify later.

3.6 Torsion-freeness and divisibility in p-groups So let us now clarify the case of torsion-free divisible groups. The divisible hull of a torsion-free subgroup A of a divisible group is unique, being (essentially) the group DA = ℚ ⊗ A. It agrees with the minimal divisible hull as defined in [101, page 233]. A torsion-free divisible group is a ℚ-vector space. We may and shall consider A as a subgroup of ℚ⊗A via the injection a 󳨃→ 1 ⊗ a : A → ℚ ⊗ A. If A is a locally compact abelian torsion-free group, we shall consider DA as the unique locally compact torsion-free divisible group ℚ ⊗ A containing A as an open subgroup. Definition 3.49. A locally compact abelian group D which is isomorphic to the divisible hull ∞

1 J ⋅ℤ pn p n=0

DℤJ = ℚ ⊗ ℤJp = ⋃ p

of ℤJp in ℚJp for a set J of cardinality ℵ will be called a locally compact abelian torsion-free divisible p-group of p-rank ℵ. The latter is identical to rankp (C) for any open compact subgroup. We briefly recall that the weight w(X) of a topological space X is the smallest cardinal α such that there is a basis ℬ of the topology of X such that α = card(ℬ). For a discrete space X we note w(X) = card(X). As we shall observe in greater detail in Theorem 3.67, we note the following preliminarily. Remark 3.50. Every divisible torsion-free abelian p-group G is algebraically a ℚp -vector space, and q 󳨃→ q⋅g : ℚp → ℚp ⋅g is an isomorphism of locally compact abelian p-groups.

3.6 Torsion-freeness and divisibility in p-groups | 73

Theorem 3.51 (Main theorem on divisible torsion-free locally compact abelian p-groups). (i) Any locally compact abelian torsion-free p-group G is an open subgroup of its divisible hull DG = ℚ ⊗ G which is a torsion-free divisible p-group of p-rank ℵ for some cardinal ℵ. Moreover, DG = ℚ⊗C for any compact open subgroup C of G with C ≅ ℤℵ p and DG is a subgroup and indeed ℚp -vector subspace of ℚℵ p. (ii) If C = ℤℵ p for any infinite cardinal ℵ, then ℚ⊗C (2ℵ ) ≅ ℤ(p∞ ) C

(1)

and rankp (C) = ℵ

and

card(

ℚ⊗C ) = 2ℵ . C

(2)

(iii) The character group of a locally compact abelian torsion-free divisible p-group Δ is divisible if and only if Δ ≅ ℚnp for a nonnegative integer n. Proof. We begin by proving (i) and first state a simple observation. A nonsingleton discrete torsion-free divisible group is a rational vector space, all monothetic subgroups are isomorphic to ℤ, and it therefore cannot be a locally compact p-group. Now let G be a nonsingleton locally compact abelian torsion-free p-group. Being nondiscrete and totally disconnected, there exists a compact open subgroup C. Then C is a compact totally disconnected and torsion-free group. Its character group therefore is a discrete divisible torsion group by Corollary 8.5 of [54]. As such it is of the form ℤ(p∞ )(J) for some set J by Proposition A1.41 of [54]. Accordingly, C ≅ ℤJp . Since ℤp is 1 J J divisible by all natural numbers relatively prime to p, it follows that ⋃∞ n=0 pn ℤp = ℚ⋅ℤp

is the divisible hull in ℚJp . In the torsion-free case, divisible hulls are unique. Hence ∞ 1 1 ⋃∞ n=0 pn ⋅C = ℚ⋅C is the divisible hull DC of C and is an open subgroup of ⋃n=0 pn ⋅G = DG . But being divisible, DC is pure in DG , whence DG /DC is a torsion-free discrete group on the one hand and a p-group on the other and so is singleton. This shows DC = DG . We know that DC is a torsion-free divisible p-group of p-rank ℵ = card J, and so this applies to DG as well. This completes the proof of (i). ∞ 1 ℵ ℵ n ℵ n ℵ Proof of (ii): We recall that ℚℵ p ⊇ ℚ ⊗ C = ⋃n=0 pn ⋅ℤp . Now ℤp /p ⋅ℤp ≅ ℤ(p ) . Accordingly, 1 ⋅ℤℵ p pn ℤℵ p

≅ ℤ(pn )

ℵ

with ℤ(pn ) =

1 ⋅ℤ/ℤ. pn

Hence ∞ ℚ⊗C ℵ ℵ = ⋃ ℤ(pn ) = tor ℤ(p∞ ) . C n=0

74 | 3 Abelian periodic groups But the discrete group tor ℤ(p∞ )ℵ is a divisible p-group of cardinality 2ℵ and thus of ℵ p-rank 2ℵ . Hence it is isomorphic to ℤ(p∞ )(2 ) . This proves (1). For the first equation in (2) see Definition 3.49 The second equation is immediate from (1). ℵ Proof of (iii): By (i) we can assume that Δ = ℚ⊗ℤℵ p . Then C := ℤp is a compact open ⊥ ⊥ ̂ has the compact open subgroup C and C is isomorphic to the subgroup of Δ. Then Δ character group of Δ/C by the annihilator mechanism (see, e. g., [54], Theorem 7.64, p. 359), and thus (in view of [54], Theorem 7.76 (i)) we have w(C ⊥ ) = w(Δ/C) = 2ℵ by ̂ ⊥ ) = w(C) = ℵ. The group Δ ̂ is a torsion-free locally compact (ii) above while w(Δ/C ̂ and abelian p-group (see [54], Proposition 8.3 (ii), p. 376). Hence part (i) applies to Δ ̂ is an open subgroup of ℚ ⊗ C ⊥ . shows that Δ ̂ is divisible iff Δ ̂ = ℚ ⊗ C ⊥ , that is, iff card(Δ/C ̂ ⊥ ) = ℵ on the one hand Now Δ ⊥ 2ℵ and card(ℚ ⊗ C ) = 2 by (ii) above on the other. This is impossible if ℵ is infinite. ̂ is divisible then ℵ is a nonnegative integer n. Conversely, if ℵ = n, then Thus, if Δ ̂ ≅ ℚn is divisible. This concludes the proof. Δ = ℚ ⊗ ℤnp = ℚnp and then Δ p Corollary 3.52. Any torsion-free divisible periodic p-group G is isomorphic to I

loc

I

I

ℚ ⊗ ∏ ℤpp = ∏ (ℚ ⊗ ℤpp , ℤpp ) p prime

p prime

for a family {Ip : p prime} of sets. Proof. Let C be a compact open subgroup of G. Then as a compact torsion-free group, up to isomorphism, I

C = ∏ ℤpp p prime

for a family {Ip : p prime} of sets, since its character group is a divisible torsion group and thus is a direct sum of Prüfer groups ℤ(p∞ ). By Theorem 3.51 (i), G = ℚ ⊗ C up to isomorphism. For each prime p, the p-primary component is the divisible hull I Gp = ℚ ⊗ Cp of the p-primary component Cp = ℤpp of C. So the Braconnier primary decomposition Theorem 3.3 for periodic locally compact abelian groups proves the remainder of the corollary. k

k

Since every rational number is of the form m/p1 1 ⋅ ⋅ ⋅ pnn for some integer m and primes p1 , . . . , pn the equality of ℚ ⊗ C and the local product can be observed directly. Corollary 3.53. For a locally compact abelian torsion-free divisible p-group, the following conditions are equivalent: (1) rankp (C) is finite for every open compact subgroup C, (2) G ≅ ℚnp for some n = 0, 1, 2, . . . , (3) the scalar multiplication x 󳨃→ p⋅x is an automorphism of topological groups; equivalently, G is a topological ℚp -vector space,

3.6 Torsion-freeness and divisibility in p-groups | 75

(4) G is sigma-compact, ̂ is divisible. (5) G Proof. (1), (2), and (3) are equivalent after Example 3.43. Clearly a finite-dimensional ℚp -vector space with its natural topology is sigma-compact since ℚnp /ℤnp ≅ ℤ(p∞ )n is countable. It remains to show that G fails to be sigma-compact if its p-rank is infinite. In that case we observe that we have G/C ≅ ℚ⋅ℤJp /ℤJp ⊇

J 1 ⋅ℤ p p ℤJp

≅ ℤ(p)J ,

so that G is the disjoint union of at least 2ℵ0 copies of C if J is infinite. So G cannot be sigma-compact in this case. The equivalence of (5) and (1) was shown in Theorem 3.51 (iii). After Theorem 3.51 our earlier Corollary 3.48 takes on the following explicit form ̂ has no nondiscrete for which we recall that a profinite group C is called coreduced iff C divisible subgroup. Theorem 3.54. Let G be a divisible locally compact abelian p-group. Then we find a coreduced compact p-subgroup Cp of G and sets I and J such that G ≅ (ℚ ⊗ ℤIp ) × DCp × ℤ(p∞ ) , (J)

where DCp is the divisible hull of Cp as in Proposition 3.42. In the context of the preceding Theorem 3.51 we return briefly to Example 3.43. The group G = ℚJp is a locally compact abelian divisible torsion free group containing the divisible hull D = DG of the compact open subgroup ℤJp of G. Assume now that J is infinite. Then the containment is proper, and G is therefore, by Proposition 3.45, of the form G = D ⊕ D∗ with a discrete divisible torsion-free group D∗ ≅ G/D. Let A = ℤ(p∞ )J 1 J ∗ J and S = ( p1 ℤ/ℤ)J be its socle. Then tor A = ⋃∞ n=1 ( pn ℤ/ℤ) , and D ≅ A/ tor A ≅ ℚ . The point that we observe here is that G is not a p-group, and therefore fails to be isomorphic to the divisible hull of a compact power of groups ℤp such as, for instance, D. So we model the next example after our examples in the torsion case in order to show that a maximal divisible subgroup of a torsion-free locally compact abelian p-group need not be closed. ℕ ℕ We let Δ = ℚℕ p , D = ℚ ⊗ ℤp , and C = ℤp and we interpolate between the compact open subgroup C and its divisible hull D the p-group P = (ℚp , ℤp )loc,ℕ .

(∗)

76 | 3 Abelian periodic groups Example 3.55. The group P in (∗) is a locally compact abelian torsion-free p-group with compact open subgroup C and D/C≅ℤ(p∞ )(ℕ) . But P is not divisible since (cn )n∈ℕ ∈ C, cn = 1, for all n does not have a pth root in P. Indeed its unique pth root (1/p, 1/p, . . . ) in ℚℕ ⊆ ℚp ℕ is not contained in P. The subgroup E := ℚ(ℕ) p is a dense proper subgroup of P which is divisible. Hence the maximal divisible subgroup MP of P is dense and proper. We note that P = E + C while E ∩ C = ℤ(ℕ) p . Thus P/E ≅ C/(E ∩ C) =

ℤp ℕ

ℤ(ℕ) p

.

In Lemma 3.57 we shall show that this group is isomorphic to from abstract abelian group theory need to be re-counted.

(∗∗) ℤℕ . A few concepts ℤ(ℕ)

Remark 3.56. An abelian group is cotorsion provided every extension by some torsion-free group splits. A torsion-free cotorsion group is algebraically compact and is described by Kaplansky’s theorem (see [32, 33]). A result by Hulanicki, see [59], says that ∏i≥1 Gi / ⨁i≥1 Gi is algebraically compact whenever a sequence of abelian groups (Gi )i≥1 is given. In our case we choose Gi = ℤp . Lemma 3.57. There are algebraic isomorphisms K≅

ℵ0 ℤℕ 0 ≅ ℚ(2 ) × ∏ (ℤℵ p ). (ℕ) ℤ p prime

(∗∗∗)

Proof. Since ℤℕ p is compact and hence cotorsion and K is a homomorphic torsion-free image, K is algebraically compact (see [32, VII]). First of all K = D⊕R for D the maximal divisible subgroup and R reduced. By the torsion-freeness, using Kaplansky’s theorem one can provide a cardinal m0 for which D ≅ ℚ(m0 ) and R ≅ ∏q Aq where, for q any (m )

prime, Aq is the q-adic completion of a direct sum ℤp p . One verifies that ℤℕ /ℤ(ℕ) is a torsion-free algebraically compact pure subgroup of K with corresponding cardinal invariants n0 = c and nq = c. Then the pureness of this embedding yields estimates of cardinalities from below, i. e., c ≤ m0 and c ≤ mp . From this and the fact that P/E has cardinality c we deduce the equalities c = m0 and c = mp . Hence we have an (abstract) isomorphism K ≅ ℤℕ /ℤ(ℕ) . The structure of ℤℕ /ℤ(ℕ) follows from [32, Chapter 40].

Returning to Example 3.55 and denoting by MK the maximal divisible subgroup of K, and further U its full inverse image in P, we observe that U/E ≅ MK and since divisible subgroups split we have U = E ⊕ M 󸀠 with M 󸀠 ≅ MK , whence U = MP , the maximal divisible subgroup of P. For giving an explicit description of the maximal divisible subgroup MK of K we use the convention p∞ = 0 and say that a sequence ℓ ∈ (ℕ ∪ {∞})ℕ in ℕ ∪ {∞} has

3.6 Torsion-freeness and divisibility in p-groups | 77

finite sublevel sets iff (∀m ∈ N) |{n ∈ ℕ : ℓn ≤ m}| ≤ ∞. Now let ℕ

L = {ℓ ∈ (ℕ ∪ {∞}) : ℓ has finite sublevel sets}. The set L is a lattice in the componentwise partial order. For each ℓ ∈ L set Hℓ = pℓ1 ⋅ℤp × pℓ2 ⋅ℤp × ⋅ ⋅ ⋅ ⊆ ℤℕ p . The function ℓ 󳨃→ Hℓ from L into the lattice of subgroups of ℤℕ p is an order reversing lattice morphism. We make the following claims. (ℕ) (i) The term (Hℓ + ℤ(ℕ) p )/ℤp is divisible. For a proof we let m ∈ ℕ and z = (pℓ1 z1 , pℓ2 z2 , . . . ) ∈ Hℓ . Then F := {n ∈ ℕ : ℓn ≤ m} is finite since ℓ has finite sublevel sets. Now we define for n ∈ ̸ F,

ℓ󸀠 = ℓn , ℓ󸀠 = { n m,

for n ∈ F

and z 󸀠 = (pℓ1 z1 , pℓ2 z2 , . . . ) = pm (pℓ1 −m z1 , pℓ2 −m z2 , . . . ) = pm z 󸀠󸀠 . 󸀠

󸀠

󸀠

󸀠

But now z − pm z 󸀠󸀠 = z − z 󸀠 ∈ ℤ(ℕ) p , and this proves claim (i). Now we set H = ⋃ℓ∈L Hℓ . Since ℓ 󳨃→ Hℓ is monotone, H is a subgroup of ℤℕ p .

(ℕ) (ii) We have (H + ℤℕ p )/ℤp = MK . The containment ⊆ follows from claim (i). Conversely, assume that an element (ℕ) ℓ ℓ z = (z1 , z2 , . . . ) in ℤℕ p is divisible modulo ℤp . Then we write z = (p1 x1 , p2 x2 , . . . ) with

maximal exponents ℓm ∈ ℕ ∪ {∞}. Since z is divisible by pm for all m modulo ℤ(ℕ) p we conclude that for each m we have ℓn ≥ m with at most finitely many exceptions, that is, ℓ has finite sublevel sets and so z ∈ Hℓ ⊆ H modulo ℤ(ℕ) p . This proves claim (ii). This concludes the analysis of Example 3.55. For a full understanding of the example we recall the following remark. Remark 3.58. Let J be an arbitrary set. The group ℤp and thus also C = ℤJp is reduced, that is, does not contain any nonzero divisible subgroup (see Proposition 5.30). ℕ (ℕ) So ℤℕ p is reduced while in fact we argued that ℤp /ℤp has a large divisible subgroup.

78 | 3 Abelian periodic groups 3.6.1 Splitting in torsion-free p-groups In spite of an abundance of counterexamples, some splitting results hold in torsionfree locally compact abelian groups. Recall that any torsion-free compact p-group is isomorphic to ℤJp for some set J and that these groups are the projectives in the category of compact p-groups. This is a consequence of the fact that their divisible duals are injective in the category of discrete abelian p-groups. Lemma 3.59. Let C = ℤJp and let P be a closed pure subgroup. Then there is a closed subgroup F such that C = P ⊕ F, algebraically and topologically. Proof. The group C/P is a compact p-group which is torsion-free since P is pure. Since C/P is projective, there is a morphism j: C/P → C such that for the quotient epimorphism e: C → C/P the following diagram is commutative: C ↑ ↑ idC ↑ ↑ ↓ C

←󳨀󳨀󳨀󳨀 j

󳨀󳨀󳨀󳨀→ e

C/P ↑ ↑ ↑ ↑ idC/P ↓ C/P,

that is, e ∘ j = idC/P , saying that R = j(C/P) is a retract. Thus there is a closed subgroup R such that C = P ⊕ R in the category of compact abelian p-groups. We can reformulate this lemma as follows. Lemma 3.60. For any closed pure subgroup P of a compact torsion-free p-group C there is an endomorphism q of C such that q2 = q and P = q(C). Proposition 3.61. Any closed divisible subgroup of a divisible, torsion-free locally compact abelian p-group is a direct summand, algebraically and topologically. Proof. Let D be a divisible, torsion-free locally compact abelian group and V be a closed divisible subgroup. From Theorem 3.46 we know that there is a compact open subgroup C of D such that D may be identified with ℚ ⊗ C. The subgroup P := V ∩ C of C satisfies n⋅C ∩ P = n⋅C ∩ V = n⋅C ∩ n⋅V = n⋅(C ∩ V) = n⋅P since V is divisible and D is torsion-free, and so P is a pure subgroup of C. Also C ≅ ℤJp for some set J since C is a compact torsion-free p-group. Hence Lemma 3.59 and Lemma 3.60 apply and produce an endomorphism q: C → C such that q2 = q and q(C) = P = C ∩ V. Since D is torsion-free divisible and D = ℚ⋅C, every element d ∈ D is uniquely of the form d = p−n ⋅c for c = pn ⋅d ∈ C, and the endomorphism q of C extends uniquely to an endomorphism f : D → D by f (d) = p−n ⋅q(c). It satisfies f 2 = f . An element d󸀠 ∈ D is in V iff it is of the form d󸀠 = p−n ⋅c󸀠 with c󸀠 ∈ P. That is the case iff there is a c ∈ C such that c󸀠 = q(c) and so d󸀠 = p−n ⋅c󸀠 = p−n ⋅q(c) = f (d) for d = p−n ⋅c. This shows that f (D) = V and therefore there is a closed subgroup W of D such that D = V ⊕ W (algebraically and topologically). This proves the proposition.

3.6 Torsion-freeness and divisibility in p-groups | 79

For the category of abelian groups we know for a fact that each divisible subgroup of an abelian group is a direct summand. Now the preceding proposition enables us to prove the following fact for the category of torsion-free locally compact abelian p-groups. Theorem 3.62. Every closed divisible subgroup of a torsion-free locally compact abelian p-group is a direct summand (algebraically and topologically). Proof. Let V be a closed divisible subgroup of a torsion-free locally compact abelian p-group G. By Theorem 3.51 (i) there is a torsion-free divisible locally compact abelian hull, i. e., ∞

DG = ℚ ⊗ G = ⨁ p−n ⋅G ⊇ V. n=0

By Proposition 3.61 there is a closed subgroup W of DG such that DG = V ⊕ W in the category of locally compact abelian groups. By the modular law, since V ⊆ G we have G = V ⊕ (W ∩ G) in the category of locally compact p-groups. One may reformulate this fact as follows. Corollary 3.63. Let K be a closed divisible torsion-free subgroup of a locally compact abelian p-group G and suppose G/K is torsion-free as well. Then K is a direct summand (algebraically and topologically). We recall that Theorem 3.51 gives us the precise structure of torsion-free divisible locally compact abelian groups, and in fact this structure is determined by one cardinal. In particular, we now have the following corollary. Corollary 3.64. Any subgroup isomorphic to ℚm p for a natural number m in a torsionfree locally compact abelian p-group splits. Proof. The group ℚm p is locally compact. Any locally compact subgroup of a Hausdorff topological group is closed. Hence the corollary follows from Theorem 3.62.

3.6.2 The largest divisible subgroup We recall that every abelian group G has a unique largest divisible subgroup div(G). If G is a torsion-free group, then div(G) is a ℚ-vector space. If, however, we proceed to locally compact abelian p-groups, we note the following. Lemma 3.65. If G is a torsion-free locally compact abelian p-group, then div(G) is a ℚp -vector space. Moreover, q 󳨃→ q⋅g : ℚp → ℚp ⋅g is an isomorphism of locally compact abelian groups for each nontrivial g ∈ div(G).

80 | 3 Abelian periodic groups Proof. By Theorem 3.51 for any compact open subgroup C of G we have C ⊆ G ⊆ D, where D = ℚ ⊗ C = ℚ ⊗ G is a ℚp -vector space. For h ∈ G we have ⟨h⟩ = ℤp ⋅h and z 󳨃→ z⋅h : ℤp → ⟨h⟩ is an isomorphism of compact groups. If in addition h ∈ div(G), then Q⋅h ⊆ D is in fact a subgroup of div(G). Since ℚp = ⋃n∈ℕ ℤp ⋅ p1n , for each 0 ≠ g ∈ div(G) we have ℚp ⋅g = ⋃ ℤp ⋅( n∈ℕ

1 ⋅g) ⊆ div(G). pn

Thus div(G) is a ℚp -vector subspace of D. Moreover, since q 󳨃→ q⋅g : ℤp ⋅ p1n → ⟨ p1n ⋅g⟩ is a homeomorphism and p1n ℤp is a compact open subgroup of ℚp , the function q 󳨃→ q⋅g : ℚp → ℚp ⋅g is a homeomorphism for each nontrivial g ∈ div(G). Lemma 3.66. If G is a torsion-free locally compact abelian p-group, then div(G) is closed in G. Proof. Note that inside D = ℚ ⊗ G we have div(G) = G ∩ p1 ⋅G ∩ and that G ∩

1 ⋅G pn

1 ⋅G p2

∩

1 ⋅G p3

1 ⋅G n p n∈ℕ

∩ ⋅⋅⋅ = G ∩ ⋂

(∗)

is the inverse image of the continuous function g 󳨃→ pn ⋅g : G → G.

Hence by (∗), the subset div(G) of G is closed. Recall from [54, Definition A1.29] that an abelian group A is called reduced if div(A) = {0}. From Theorem 3.62 we now obtain the following at once. Theorem 3.67. Every torsion-free locally compact abelian p-group G is the direct sum, algebraically and topologically, of its largest divisible subgroup div(G) and a reduced closed subgroup R. Note that div(G) is fully characteristic, but that R is not unique. Still R is isomorphic to the canonical group G/ div(G). We shall see reduced torsion-free locally compact abelian p-groups with divisible torsion-free quotients below (see Example 3.81). If a group G has a divisible factor group G/C then this does not imply at all that G has nontrivial divisible subgroups: A free abelian group with countably many generators is readily mapped onto ℚ. There are some nontrivial aspects in the context of p-groups where a divisible quotient may cause some divisible subgroup to exist and to split.

3.6 Torsion-freeness and divisibility in p-groups | 81

For an illustration let us first look at a discrete example which is established via duality from profinite groups. Lemma 3.68. Let G be a compact p-group with a compact torsion-free open subgroup C of finite index. Then the following statements hold: (i) C ∩ tor(G) = {0}, (ii) there is a compact open torsion-free subgroup K such that G = K ⊕ tor(G), (iii) K contains an isomorphic copy of C. Proof. (i) Since C is torsion-free we have C ∩ tor(G) = {0}. (ii) The finiteness of G/C implies that tor(G) is finite and thus is closed. Then by Proposition 3.14 we find a compact subgroup K such that G = K ⊕tor(G). Automatically, K is open and torsion-free. (iii) The subgroup C meets tor(G) trivially. Hence the projection G → K with kernel tor(G) maps C isomorphically onto a subgroup of K. By passing to the dual we get the following counterpart. Proposition 3.69. Let G be a discrete p-group with a finite subgroup F such that G/F is divisible. Then the following statements hold: (i) F + div(G) = G, (ii) there is a finite subgroup E such that G = div(G) ⊕ E, (iii) E is a quotient group of F. ̂ is compact and we apply the annihilator mechanism of duality Proof. The dual G (cf. [54], Theorem 7.64 (vi)) to find that the compact open subgroup F ⊥ is such that ̂ ⊥ , being the character group of F, is finite. So G ̂ satisfies the hypotheses of LemG/F ⊥ ̂ = {0}. Thus F + tor(G) ̂ ⊥ = G. The ma 3.68. So part (i) of that lemma says that F ∩ tor(G) ⊥ ̂ annihilator tor(G) is Div(G), the set of all divisible elements g ∈ G, that is, the set of all those g ∈ G such that (∀n ∈ ℕ)(∃x ∈ G) pn ⋅x = g (see, e. g., [54] Propositions 8.2 and 8.3 (ii)). Thus F + Div(G) = G. ̂ Lemma 3.68 (ii) assures the existence of a subgroup K ≅ ℤℵ p of G for some cardinal ̂ = K ⊕ tor(G). ̂ By duality again we conclude G = (tor(G)) ̂ ⊥ ⊕ K ⊥ , where ℵ such that G ⊥ ∞ (ℵ) ̂ (tor(G)) ≅ ℤ(p ) has finite index and thus must be div(G). Thus, there is a finite subgroup E = K ⊥ such that G = div(G) ⊕ E, which proves (ii). Moreover, since div(G) ⊆ Div(G) we must have equality here, and so (i) is proved as well. Finally, F/(F ∩ div(G)) ≅ G/ div(G) ≅ E in view of (i) and (ii), and this proves (iii) and this completes the proof of the lemma. The long and the short of Proposition 3.69 is that in an abstract p-group G which is divisible modulo a finite subgroup, div(G) has finite index. We shall now pursue this issue for torsion-free locally compact abelian p-groups.

82 | 3 Abelian periodic groups Proposition 3.70. Let G be a locally compact abelian p-group such that, for some compact open subgroup C, we have G/C ≅ ℤ(p∞ ). Then div(G) ≠ {0}. ∞

Proof. Since ℚ/ℤ = ∏p󸀠 prime ℤ(p󸀠 ) we have a surjective morphism f : ℚ → G/C. Then by Theorem 1.34 f lifts to a nondegenerate morphism F: ℚ → G. This proves the claim. We showed in Theorem 3.67 that in every locally compact abelian torsion-free group G the unique fully characteristic closed largest divisible subgroup div(G) satisfies G = R ⊕ div(G). But this does not yet show how div(G) relates to any of the commensurable compact open subgroups C. The following corollary contributes information to the structure of the reduced group R and the presence of compact open subgroups C. Corollary 3.71. Let G be a reduced locally compact abelian p-group and C be a compact open subgroup. Then G/C is a reduced group. Proof. Proof by contradiction: Suppose that the claim is false. Recall that G/C is a discrete p-group and notice that then there is a closed subgroup H ⊆ G containing C such that H/C ≅ ℤ(p∞ ). But then div(H) ≠ {0} by Proposition 3.70, contradicting the hypothesis. The following theorem exploits this information; remarkably, despite all of the information we have accumulated earlier on torsion-free locally compact p-groups, it is now all but trivial. Theorem 3.72. Let G be a torsion-free locally compact abelian p-group. Assume that there is a compact open subgroup C such that the discrete factor group G/C is divisible. Then there is a profinite subgroup R such that, algebraically and topologically, G = R ⊕ div(G). Proof. By Theorem 3.67 there is an algebraic and topological direct sum representation G = R ⊕ div(G). If C is a compact open subgroup such that G/C is divisible, then the homomorphic image G G/C ≅ C + div(G) (C + div(G))/C is divisible as well. But that is a homomorphic image of the reduced group G/ div(G) ≅ R. By Corollary 3.71 this implies C + div(G) = G. Then R ≅ G/ div(G) ≅ C/(C ∩ div(G)) is compact, which we had to show. Example 7.42 shows that one cannot dispense with the torsion-freeness of G. Corollary 3.73. If C ∗ ⊆ C is defined as C ∗ = R ∩ C, then C ∗ ⊕ (C ∩ div(G)) has finite index in C, further C ∗ ⊕ div(G) has finite index in G.

3.7 Dense divisible subgroups | 83

If C ∗∗ ⊇ C is defined as C ∗∗ = C + R, then C ∗∗ is a compact open subgroup of G and C has finite index in C ∗∗ and G = C ∗∗ + div(G). Proof. The subgroup C ∗ := C ∩ R ⊕ (C ∩ div(G)) is a compact open subgroup of C and then also of G; so C ∗ has finite index in C and C ∗ ⊕ div(G) has finite index in G. Also, C ∗∗ := C + R is a compact open subgroup and C ∗∗ /C is finite, while G = R ⊕ div(G) ⊆ C ∗∗ + div(G).

3.7 Dense divisible subgroups It is clearly useful to have information on a locally compact abelian p-group G with a dense divisible subgroup D and that, in spite of examples like Example 3.55, we may get such information provided we allow certain additional conditions to be satisfied. We recall from Definition 3.27 that a topological abelian group with a dense divisible subgroup is called densely divisible. To begin with, the following lemma is still quite general. Lemma 3.74. Let G be a densely divisible locally compact abelian p-group and let C be a compact open subgroup. Then the following statements hold: ̂ is a torsion-free locally compact abelian p-group so that the an(i) The dual group G ⊥ ̂ and C ⊥ ≅ ̂ is a compact open subgroup such that G/C ̂ ⊥ ≅ C nihilator C of C in G (G/C) ̂ , (ii) G/C is a discrete divisible p-group and therefore is a direct sum of copies of the Prüfer group ℤ(p∞ ); dually, C ⊥ ≅ ℤℵ p for some cardinal ℵ, ̂ (iii) G is an open subgroup of some divisible torsion-free locally compact abelian p-group Δ ≅ ℚ ⊗ C⊥ , ̂ modulo a compact kernel. (iv) G is isomorphic to a quotient group of Δ Proof. (i) Let D be a dense divisible subgroup of G. Then p⋅G ⊇ p⋅D = D is dense in G, and so the endomorphism x 󳨃→ p⋅x : G → G is an epimorphism in the category ̂ → G ̂ is a of locally compact groups. Then the adjoint endomorphism χ 󳨃→ p⋅χ : G monomorphism in that category, which means that it is injective. Thus m⋅χ = 0 implies ̂ χ = 0 since this is true for m = p and, as in any p-group, for (m, p) = 1 as well. Hence G ⊥ ̂ are consequences of the is torsion-free. The remainder of the assertions on C and C annihilator mechanism of Pontryagin duality (see, e. g., [54], Theorem 7.64). (ii) Since D is dense in G, (D + C)/C is dense in G/C. But C is open in G and so G/C is discrete, showing that (D + C)/C = G/C. Thus G/C is a discrete divisible p-group and thus is a direct sum of Prüfer groups by the structure theory of (abstract) abelian groups. (See, e. g., [54], Proposition A1.41.) The remainder follows via duality. (iii) This is an immediate consequence of (i), (ii), and Theorem 3.51 (i).

84 | 3 Abelian periodic groups ̂ → ℚ ⊗ C ⊥ → A → 0 is exact with a discrete A, then (iv) If 0 → G ̂→G→0 0 󳨃→ A⊥ → Δ is exact with a compact A⊥ by duality again. A further systematic use of duality allows us now to conclude the following information. Theorem 3.75. Let G be a densely divisible locally compact abelian p-group and C a compact open subgroup such that at least one of the following conditions is satisfied: (a) The discrete p-group G/C has finite p-rank m. (b) The compact p-group C has finite p-rank m. Then G is divisible. In case (a), G is a quotient of ℚm p and the p-rank of any of its open compact subgroups is ≤ m. ∞ (ℵ) In case (b), one has G ≅ ℚm for some cardinal ℵ. p ⊕ ℤ(p ) Proof. Assume case (a). The claim will follow at once from Lemma 3.74 as soon as we ̂ ≅ ℚm , and since ℚm is isomorphic to its character group, this will can establish that Δ p p follow from ℚ ⊗ C ⊥ ≅ ℚm p.

(c)

Since Lemma 3.74, (i) says that C ⊥ is a compact torsion-free abelian group; we know that it is ≅ ℤIp for some set I. Now Lemma 3.74 (iii) implies that (up to isomorphism and identifying C ⊥ with ℤIp ) ̂ ⊆ ℚ ⊗ ℤI . ℤIp ⊆ G p

(d) Now Condition (a) says that

m = rankp G/C = rankp (G/C) ̂ = rankp (C ⊥ ) = rankp (ℤIp ) = card(I). m Thus Δ ≅ ℚ ⊗ ℤm p = ℚp , and this proves the assertion in case (a). Now we assume case (b). Then the compact open abelian group C of finite p-rank is isomorphic to ℤkp ⊕F for a natural number k (see Proposition 3.9) and a finite abelian

group F of p-rank ℓ such that k + ℓ = m. Then the summand ≅ ℤkp is commensurable to C and thus satisfies the hypotheses spelled out for C. Hence in order to simplify notation we may assume that C ≅ ℤm p . Then ̂ I = G/C ̂ ⊥ ≅ ℤ(p∞ )m . G/ℤ p

(∗)

̂ = R ⊕ div G ̂ for a profinite torsionAt this point we apply Theorem 3.72 and find that G ⊥ ⊥ free p-group R. The compact open subgroups C and R + C are commensurable, and

3.7 Dense divisible subgroups | 85

since we may replace C by a smaller compact open subgroup of G, we may replace C ⊥ by the possibly larger subgroup R + C ⊥ without loss of generality; therefore we may ̂ = C ⊥ + div(G). ̂ (Cf. Corollary 3.73.) Then from (∗) we assume that R ⊆ C ⊥ and thus G have ̂ div(G)

̂ C ⊥ ∩ div(G)

≅

̂ ̂ C ⊥ + div(G) G m = ⊥ ≅ ℤ(p∞ ) . ⊥ C C

(∗∗)

̂ and thus Let S̃ be the inverse image of the socle of ℤ(p∞ ). Then pS̃ = C ⊥ ∩ div(G) m ⊥ ̃ S̃ ≅ ℤ(p) . Therefore C ∩div(G) ̂ has p-rank m. By Lemma 3.65 div(G) ̂ is a ℚ -vector S/p p ̂ space and by our observations the latter has dimension m. Thus div(G) ≅ ℚm p. ̂ ≅ ℚm is ℚm and since the dual of R ≅ ℤℵ for some cardinal Then the dual of div(G) p

ℵ is ≅ ℤ(p∞ )(ℵ) , in the end we have

p

∞ G ≅ ℚm p ⊕ ℤ(p )

p

(ℵ)

.

Therefore, G is divisible as asserted. One obtains from this [3, Theorem 2] for locally compact abelian p-groups. In view of Theorem 3.75 one is faced with the following general situation: If G is any locally compact abelian p-group, the largest divisible subgroup div(G) exists as for any abstract abelian group. Our examples show that it may not be closed. The closure div(G), however, is a densely divisible abelian p-group, and our discussion of densely divisible locally compact abelian p-groups applies. If a locally compact abelian p-group A contains a dense divisible subgroup D, we cannot say much about the structure of D as an abstract abelian group; for instance, the cyclic subgroup ℤ⋅d generated by d ∈ D is either a cyclic group of p-power order or it may be algebraically isomorphic to ℤ. It is therefore not particularly meaningful to speak of the p-rank of a dense subgroup of A as the example A = ℚp shows at once. At any rate, the group A has compact open subgroups C, and so one has available the compact group C ∩ D and the discrete group D/(C ∩ D) ≅ (C + D)/C with either of these available for the hypothesis of finite rank, so that Theorem 3.75 becomes available showing that D ⊆ div(G) and giving the precise structure of D. These remarks apply, in particular, to D = div(G) itself. Here is an example of such a reasoning. Proposition 3.76. Let G be a locally compact abelian p-group and assume that for one of its compact open subgroups C the p-rank of div(G) ∩ C is finite. Then div(G) is closed. If div(G) ∩ C is a direct summand of C algebraically and topologically then there is a direct decomposition algebraically and topologically G = R ⊕ div(G) for some closed reduced subgroup R of G.

86 | 3 Abelian periodic groups Proof. We apply Theorem 3.75 (b) to div(G) and conclude that div(G) is closed and isomorphic to ℚm p ⊕T for an m ∈ ℕ0 with a discrete divisible torsion group T = tor(div(G)). So T ∩ C is finite, and we may replace C by a smaller open compact subgroup such that T ∩ C = {0}. Then T is a discrete and divisible subgroup of G, and so Lemma 3.13 shows that there is a closed subgroup S ⊆ G such that G = tor(div(G)) ⊕ S algebraically and topologically. Thus from now on we may assume that T = {0} and so div(G) ≅ ℚm p. m Therefore we may assume from now on that C ∩ div(G) ≅ ℤp is a direct summand of the compact p-group C, that is, C = (div(G) ∩ C) ⊕ L, algebraically and topologically. By the injectivity of div(G) in the category of abelian groups, the projection e: div(G) ⊕ L → div(G) extends to a projection pr: G → div(G). Since C is open in G and the restriction of pr to C is continuous, pr is continuous. Hence G = div(G) ⊕ R where R := ker(pr) provides the desired algebraic and topological decomposition of G. Remark 3.77. The preceding result compares with Theorem 2.2 in [94], which states, a bit reformulated, that a locally compact abelian p-group G with a closed divisible sigma-compact subgroup D and factor group G/D torsion decomposes algebraically and topologically as direct sum G = D ⊕ S for a closed subgroup S of G. Corollary 3.22 discusses an example of a locally compact abelian p-group G such that div(G) ≅ ℚp is closed and of p-rank 1 while it does not split as a direct summand of G topologically.

3.8 Nonsplitting of ℚp in the presence of torsion An example where G/ tor(G) ≅ ℚp does not split has been given in Section 3.3.1. Here we shall exhibit in Example 3.81 a torsion-free locally compact abelian p-group with a quotient that is isomorphic to ℚp but which does not split, and in Theorem 3.84 we shall find a locally compact abelian and sigma-compact p-group with a closed subgroup isomorphic to ℚp that does not split. These examples are in contrast with Theorem 3.62 and the subsequent corollary, as they show that the assumption of torsionfreeness cannot be dismissed. We recall ℕ0 = {0, 1, 2, . . . }.

3.8 Nonsplitting of ℚp in the presence of torsion

| 87

Lemma 3.78. We let k: ℤ(p∞ ) → ℤ(p∞ )(ℕ0 ) be defined by k(x) := (x, p⋅x, p2 ⋅x, p3 ⋅x, . . . ) ̂ ℤℕ0 → ℤ be the dual morphism, where we identify ℤ with the character and let κ = k: p p p ℕ group of ℤ(p∞ ) and ℤp 0 with the character group of ℤ(p∞ )(ℕ0 ) . Then (i) k is a well defined injective morphism, (ii) κ is a surjective morphism given explicitly by ∞

κ((z0 , z1 , z2 , . . . )) = ∑ zn pn . n=0

Proof. (i) This is immediate due to the fact that each element z in the Prüfer group ℤ(p∞ ) has finite order. (ii) As a consequence of the duality between discrete and compact abelian groups, κ is surjective since k is injective. We may identify the module action (z, x) 󳨃→ z⋅x : ℤp × ℤ(p∞ ) → p1∞ ⋅ℤ/ℤ → ℝ/ℤ and the bilinear map ∞

((z0 , z1 , . . . ), (x0 , x1 , . . . )) 󳨃→ ∑ zn ⋅xn , n=0

ℤNp 0

∞ (ℕ0 )

× ℤ(p )

→ ℤ(p∞ ) ⊆ ℝ/ℤ (ℕ )

as the dual pairings of duality. Then for z = (z0 , z1 , . . . ) ∈ ℤp 0 and x ∈ ℤ(p∞ ) we n 2 have κ(z)⋅x = ∑∞ n=0 p zn ⋅x on the one hand and z⋅k(x) = (z0 , z1 , z2 . . . )⋅(x, p⋅x, p ⋅x, . . . ) = ∞ n ∑n=0 zn ⋅(p ⋅x) on the other. The right-hand sides agree, and this shows that k and κ are adjoint under the duality. Lemma 3.79. Let C = ∏n∈ℕ0 p2n ℤp ≅ ℤp 0 . There is a morphism η: C → ℤp defined by ℕ

∞

η((x0 , x1 , . . . )) = ∑ xn p−n . n=0

Proof. By the definition of C, for each n = 0, 1, . . . there is a yn such that xn = p2n yn . n −n = ∑∞ Therefore ∑∞ n=0 yn p , which converges in ℤp so that η is well defined. Let n=0 xn p ℕ0 α: ℤp → C be the isomorphism given by α((y0 , y1 , . . . )) = (y0 , p2 y1 , . . . , p2n yn , . . . ) = (x0 , x1 , . . . ). Then ∞

(η ∘ α)((y0 , y1 , . . . ))=η((x0 , x1 , . . . ))= ∑ yn pn =κ((y0 , y1 , . . . )), n=0

that is, η = κ ∘ α−1 . We saw in Lemma 3.78 that κ is a morphism and so η is a morphism.

88 | 3 Abelian periodic groups Lemma 3.80. Let G be the torsion-free locally compact abelian group loc

∏ (ℤp , p2n ℤp )

n∈ℕ0

and let C be as in Lemma 3.79. Then the morphism η: C → ℤp of Lemma 3.79 extends to ̃ : G → ℚp . a continuous open surjective morphism η (ℕ )

ℕ

Proof. (a) We let S = ℤp 0 ⊆ ℤp 0 . Then G = C + S. Now η󸀠 : S → ℚp by η󸀠 ((z0 , z1 , . . . , zn , . . . )) = ∑n∈ℕ0 zn p−n ∈ ℚp , as a finite sum, is a well-defined algebraic homomorphism. On C ∩ S the definitions of η and η󸀠 agree. (b) Now we define η∗ : C×S → ℚp by η∗ (c, s) = η(c)−η󸀠 (s). We also have a surjective morphism δ: C × S → G defined by δ(c, s) = c − s. Then ker δ = {(c, c) : c ∈ C ∩ S}, and by ̃ : G → ℚp such part (a) of the proof, η∗ vanishes on ker δ. Hence there is a morphism η ̃ ∘ δ. Moreover, for c ∈ C we have η ̃ (c) = η ̃ (δ(c, 0)) = η∗ (c, 0) = η(c), that that η∗ = η ̃ extends η which is a continuous open morphism of the open subgroup C of G. is, η ̃ is a continuous open and surjective morphism. Therefore, η ̃ : G → ℚp with Example 3.81. (i) The quotient morphism η loc

G = ∏ (ℤp , p2n ℤp ) n∈ℕ0

(∗)

constructed in Lemma 3.80 does not split, that is, there is no morphism f : ℚp → G ̃ ∘ f = idℚp . such that η (ii) The group G is a locally compact, sigma-compact, torsion-free p-group such that pG ≠ G. In fact, there is a compact open subgroup C ≅ ℤℕ p such that G/C is a (discrete!) countable torsion group of infinite exponent. ℕ

Proof. (i) The subgroup C ≅ ℤp 0 is reduced; indeed, if D were a divisible subgroup it would have to be contained in pk C since C/pk C is bounded and thus reduced. More2n over, the factor group G/C ≅ ⨁∞ n=0 ℤ(p ) is reduced as well. Hence G is reduced and so a splitting morphism f cannot exist. (ii) The assertions in (ii) are straightforward. We now wish to record the dual situation of Example 3.81 and for this purpose we record Braconnier’s theorem on the dual of a local product of locally compact abelian groups. (See [12], Theorem 1, p. 10.) Lemma 3.82. Let {(Aj , Bj ) : j ∈ J} be a family of locally compact abelian groups Aj with compact open subgroups Bj . Then the dual group of loc

G = ∏(Aj , Bj ) j∈J

(1)

3.8 Nonsplitting of ℚp in the presence of torsion

| 89

may be identified with loc

̂ = ∏(A ̂j , B⊥ ), G j

(2)

j∈J

̂ where, as usual, B⊥ j is the annihilator of Bj in Aj . ̂ may be Lemma 3.83. Let G be as in (∗) in Example 3.81. Then the character group G identified with the group G described as follows: loc

G = ∏ (ℚp /ℤp , p−2n ℤp /ℤp ).

(∗∗)

n∈ℕ0

Then G is a locally compact abelian p-group isomorphic to loc

∏ (ℤ(p∞ ), ℤ(p2n )),

n∈ℕ0

where we write ℤ(pk ) =

1 ⋅ℤ/ℤ pk

⊆

1 ⋅ℤ/ℤ p∞

= ℤ(p∞ ).

Proof. We have to show that we have a dual pairing of the groups G and G in (∗) and (∗∗), respectively. For each n ∈ ℕ0 we have a pairing ⟨−, −⟩, i. e., (z, q + ℤp ) 󳨃→ zq + ℤp : ℤp × ℚp /ℤp → ℚp /ℤp ≅

1 ⋅ℤ/ℤ ⊆ ℝ/ℤ, p∞

such that p2n ℤp and p−2n ℤp /ℤp are annihilators of each other. Fix n ∈ ℕ; we determine for which m ∈ ℕ the group p2m ℤp is annihilated by p−2n ℤn . An element z ∈ ℤp is in p2m ℤp for an m ∈ ℤ iff there is an x ∈ ℤ×p , the group of units of ℤp , such that z = p2m x; similarly an element q + ℤp ∈ ℚp /ℤp is in p−2n ℤp /ℤp iff there is a y ∈ ℤ×p such that q = p−2n y. Now the relation ⟨z, q + ℤp ⟩ = zq + ℤp = 0 holds iff p2m x⋅p−2n y = zq ∈ ℤp iff m − n ≥ 0, that is, m ≥ n. That proves that p2n ℤp is indeed the annihilator of p−2n ℤp . Then Lemma 3.82 completes the proof of the lemma. We are now set to get a very significant example by dualizing what we obtained in Example 3.81, obtaining the following noteworthy result. −2n Theorem 3.84. The group G = ∏loc ℤp /ℤp ) is a locally compact and n∈ℕ0 (ℚp /ℤp , p sigma-compact p-group with a closed subgroup

Q := {(

q + ℤp ) : q ∈ ℚp } pn n∈ℕ0

isomorphic to ℚp which is not a direct summand in the category of locally compact abelian groups.

90 | 3 Abelian periodic groups Proof. The character group of ℚp may be identified with ℚp under the dual pairing ι

(r, s) 󳨃→ rs + ℤp : ℚp × ℚp → ℚp /ℤp 󳨀󳨀󳨀󳨀→ ℝ/ℤ, where ι is the embedding morphism ≅

ℚp /ℤp 󳨀󳨀󳨀󳨀→

incl 1 ⋅ℤ/ℤ 󳨀󳨀󳨀󳨀→ ℝ/ℤ. p∞

̃ : G → ℚp for which we now obtain In Lemma 3.80 we had a quotient morphism η a dual injection ι: ℚp → G so that (∀q ∈ ℚp , (z0 , z1 , . . . ) ∈ G) ∞

⟨ι(q), (z0 , z1 , . . . )⟩ = ⟨q, η((z0 , z1 , . . . ))⟩ = ∑ qzn p−n . n=0

(3.6)

(Recall here that almost all zn ∈ ℤp are of the form zn = p2n xn for some xn so that almost all summands of the infinite series in equation (3.6) read qxn pn . Thus the convergence of the series in ℤp is never in question.) We now easily verify that equation (3.6) is satisfied if and only if loc

ι(q) ∈ G = ∏ (ℚp /ℤp , p−2n ℤp /ℤp ) n∈ℕ0

is of the form ι(q) = (q + ℤp , qp−1 + ℤp , qp−2 + ℤp , . . . ) = (

q + ℤp ) . pn n∈ℕ0

This completes the proof. We note that any subgroup that is isomorphic to ℚp is divisible and thus is a direct summand in the category of abelian groups. We also remark explicitly that the group G of Theorem 3.84 itself is not divisible. Lemma 3.85. The group G is not divisible, but p⋅G is dense in G. Proof. Let C = ∏n∈ℕ0 p−2n ℤp /ℤp ≅ ∏n∈ℕ ℤ(p2n ). Then S := C/pC ≅ ℤ(p)ℕ and we consider the quotient loc,ℕ

H := G/pC ≅ (ℤ(p∞ ), ℤ(p))

.

This group has the subgroups E = ℤ(p∞ )(ℕ) and S = ℤ(p)ℕ so that H = E + S and so H/E ≅ S/(S ∩ E) ≅ ℤ(p)ℕ /ℤ(p)(ℕ) . This group is a quotient of a GF(p)-vector space of dimension 2ℵ0 modulo a vector subspace of dimension ℵ0 and thus is isomorphic to ℤ(p)ℕ . So we have seen that G has a nondivisible quotient and therefore is nondivisible. The character group G of G is torsion-free. This implies that μGp is injective, and so G μp = (μGp ) ̂ has a dense image.

3.9 Divisible torsion groups | 91

Accordingly, the maximal divisible subgroup D = pD is a proper subgroup and it contains the torsion-free divisible subgroup Q and the dense divisible torsion subgroup (ℚp /ℤp )(ℕ) ; consequently it is dense. The example must be seen in contrast with the positive results in the torsion-free case of p-groups as exemplified in Theorem 3.62. The vector space splitting theorem for locally compact abelian groups, see Theorem 7.57 in [54], simply fails to have a general p-adic analog. The situation will improve, however, when we shall restrict our attention to locally compact abelian p-groups of finite rank in Section 3.11.

3.9 Divisible torsion groups Staying within the class of divisible p-groups we leave the class of torsion-free groups and consider, by contrast, the class of locally compact abelian torsion groups. In any locally compact abelian torsion p-group G we find a compact open torsion subgroup C. Then C is coreduced, and we can apply Corollary 3.48 and obtain (in view of the fact that a discrete divisible p-group is a sum of Prüfer groups) the following. Lemma 3.86. Let G be a divisible locally compact abelian torsion p-group and C a compact open subgroup. Then, for a divisible hull DC of C and some set I, (I)

G ≅ DC × ℤ(p∞ ) .

(1)

From Proposition 3.11 (2) we get the existence of a natural number n and a finite sequence of sets I1 , . . . , In such that I

C ≅ ℤ(p)I1 × ⋅ ⋅ ⋅ × ℤ(pn ) n .

(2)

In order to calculate DC we need to determine the divisible hull of each product Ck = ℤ(pk )Ik . We consider Ck as a subgroup of the divisible group Pk = ℤ(p∞ )Ik . Each scalar multiplication x 󳨃→ pn ⋅x : Pk → Pk is surjective. So the divisible hull of Ck inside Pk is ∞

DCk = ⋃ p−m ⋅Ck .

(3)

m=0

From (2) and (3) we recognize the divisible hull of C inside ∏nk=1 Pk as ∞

DC = DC1 × ⋅ ⋅ ⋅ × DCn = ⋃ p−m ⋅C.

(4)

m=0

We may assume that the Ik are mutually disjoint. Let N := {k ∈ {1, . . . , n}|Ik ≠ 0} and set I = ⋃ Ik . k∈N

92 | 3 Abelian periodic groups Then S(C) = S(C1 × ⋅ ⋅ ⋅ × Cn ) = S(C1 ) × ⋅ ⋅ ⋅ × S(Cn ), S(C) = ℤ(p)I1 × ⋅ ⋅ ⋅ × ℤ(p)In = ℤ(p)I .

(5)

It follows that x ∈ p−m C iff pm ⋅x ∈ C; if x = (x1 , . . . , xn ) with xk ∈ Ck , then this is the case iff pm .xk ∈ Ck for all k = 1, . . . , n iff pm+k−1 ⋅xk ∈ S(Ck ) and so if pm ⋅x ∈ C then pm+n ⋅x ∈ S(C), that is, p−m C ⊆ pm+n S(C). The converse containment is trivial. Therefore, within ℤ(p∞ )I we have ∞

DC = ⋃ p−m S(C) = DS(C) .

(6)

m=0

This completes the proof of the following classification of divisible locally compact abelian torsion p-groups. Recall that we canonically identify ℤ(pk ) = p1k ⋅ℤ/ℤ with a unique subgroup of ℤ(p∞ ) = p1∞ ⋅ℤ/ℤ. For a set I and a bounded function b: I → ℕ we define U(I, b) = ∏ ℤ(pb(j) ) ⊆ ℤ(p∞ )

I

j∈I ∞

I

and

I

Δ(I, b) = ⋃ ℤ(pn ) ⊆ ℤ(p∞ ) . n=1

Then U(I, b) ⊆ Δ(I, b) and Δ(I, b) has a unique periodic locally compact abelian group topology for which U(I, b) is a compact open subgroup with its product topology. Theorem 3.87. For each pair of sets I and J and each bounded function b: I → ℕ, the locally compact abelian p-group (J)

Δ(I, b) × ℤ(p∞ )

with the discrete topology on the second factor is a locally compact abelian divisible torsion p-group, and each divisible torsion locally compact abelian p-group is so obtained. From Theorem 3.54 we get the following corollary. Corollary 3.88. Let G be a divisible locally compact abelian p-group with a closed torsion subgroup. Then G ≅ (G/ tor(G)) × tor(G) ≅ (ℚ ⊗ ℤIp ) × Δ(J, b) × ℤ(p∞ )

(K)

for suitable sets I, J, and K and some bounded function b: J → ℕ.

3.10 The p-rank of a locally compact abelian p-group

| 93

Proof. Applying Theorem 3.54 to the closed torsion subgroup tor(G) a direct decomposition tor(G) ≅ tor(DCp ) × ℤ(p(J) ) results for DCp the divisible hull of a coreduced subgroup Cp of G. Since tor(DCp ) = DCp ∩ tor(G) is closed, Proposition 3.14 provides us with a direct algebraic and topological decomposition Cp ≅ tor(Cp ) × L, where L ≅ ℤm p for some cardinal m. Therefore the bilinearity of the tensor product implies DCp = ℚ ⊗ Cp ≅ (ℚ ⊗ tor Cp ) × (ℚ ⊗ L). In particular this shows that tor(DCp ) = ℚ ⊗ tor Cp splits. Theorem 3.54 gives rise to the direct decomposition G ≅ (ℚ ⊗ ℤIp ) × tor(DCp ) × (DCp / tor(DCp )) × ℤ(p(J) ). Hence tor(G) splits. The result follows from applying Theorems 3.51 and 3.87.

3.10 The p-rank of a locally compact abelian p-group It is overdue by now that we should propose a theory of p-rank applicable to arbitrary locally compact abelian p-groups (and then to all periodic groups). So let G be a locally compact abelian p-group and let C be any compact open subgroup. The discourse in the preliminary Section 3.2 secures in all clarity the definitions of ̂ ⊥ ) and rank (G/C). If K is any other compact open subgroup of rankp (C) = rankp (G/C p G, then C and K are commensurable, and then it is a straightforward exercise using Remark 3.8 to show that rankp (C) is finite iff rankp (K) is finite and rankp (G/C) is finite iff rankp (G/K) is finite. Also, if rankp (C) is infinite, then rankp C = rankp K, and, similarly, if rankp (G/C) is infinite, then rankp (G/C) = rankp (G/K). Definition 3.89. Let G be a locally compact abelian p-group and let C be any compact open subgroup. We define the p-rank of the pair (G, C) to be rankp (G, C) = max{rankp C, rankp (G/C)}. Finally, let 𝒩 (G) denote the lattice of all compact open subgroups C of G. Then we define the p-rank of G to be rankp G = sup{rankp (G, C) : C ∈ 𝒩 (G)}.

94 | 3 Abelian periodic groups We note that G is of finite p-rank iff both rankp C and rankp (G/C) are finite. If C is a compact open subgroup of G then there are two possibilities: (a) rankp (C) is finite and (b) rankp (C) is infinite. In case (a), we have C ≅ ℤm p ⊕ F for a nonnegative integer m and a finite group F by Proposition 3.9. As a consequence, all sufficiently small compact open subgroups K of C satisfy rankp (K) = m. In case (b), any compact open subgroup K of C being commensurable with C satisfies card(C/K) < ∞. Then (b) implies rankp K = rankp C. On the other hand, if K is any compact open subgroup of C, then there is an exact sequence of discrete abelian groups 0 → C/K → G/K → G/C → 0 with a finite group C/K. In case (a), if K is sufficiently small, both G/C and G/K have rank m ∈ ℕ0 , and in case (b), again G/C and G/K have the same rank. We say that a net (xj )j∈J in a partially ordered set is finally constant iff there is an index j0 ∈ J such that j0 ≤ j, k implies xj = xk . Recalling that 𝒩 (G) denotes the filter basis of all compact open subgroups of a locally compact abelian periodic abelian group G we summarize. Remark 3.90. Let G be a locally compact abelian p-group. Then the net of cardinals (rankp (G, C))C∈𝒩 (G) is finally constant. Equivalently, there is a compact open subgroup Cs such that for all compact open subgroups K ⊆ Cs we have rankp G = max{rankp K, rankp (G/K)}

= lim max{rankp C, rankp (G/C)}. C∈𝒩 (G)

We shall call a compact open subgroup such as Cs to be sufficiently small. Recalling that a compact abelian p-group of p-rank n is generated by n elements. Lemma 3.91. If G is a locally compact abelian p-group, then a natural number n is rankp G if and only if n is the smallest natural number such that every finitely generated subgroup of G is generated by at most n elements. The concept of a general p-rank on the full category of locally compact abelian p-groups is more intricate than one might expect. A few examples in the beginning may illustrate the concept. (a) Let G = ℤ(p2 )ℕ with that locally compact topology for which C = p⋅G = (p⋅ℤ(p2 ))ℕ is a compact open subgroup. Then rankp C = rankp ℤ(p)ℕ = card ℕ = ℵ0 , while rankp (G/C) = rankp ℤ(p)(2

ℕ

)

= 2ℵ0 .

3.10 The p-rank of a locally compact abelian p-group

| 95

Hence rankp (G, C) = 2ℵ0 = rankp G. ̂ = rank ℤ(p∞ ) = 1 and (b) Let G = ℚp ; we take C = ℤp . Then rankp C = rankp C p ∞ rankp G/C = rankp ℚp /ℤp = rankp ℤ(p ) = 1. Hence rankp (ℚp , C) = rankp ℚp = 1. Also rankp ℚm p = m for any natural number m. (c) Let G be a locally compact abelian torsion-free divisible group ℚ ⊗ ℤℵ p of an inℵ finite local weight ℵ. We take C = ℤp . Then rankp (C) = ℵ and rankp (G/C) = ℵ

rankp ℤ(p∞ )(2 ) = 2ℵ . Hence rankp (G, C) = 2ℵ = rankp G. This may be somewhat unexpected. The smallest infinite p-rank of any divisible torsion-free locally compact abelian p-group is 2ℵ0 . There is no divisible torsion-free locally compact abelian p-group of p-rank ℵ0 . (d) Let G = ℤp ⊕ ℤ(p). Set C1 = ℤp , C2 = pℤp . Then rankp (G, C1 ) = 1, rankp (G, C2 ) = 2, and rankp G = 2. It is straightforward to verify that the general definition of rank in Definition 3.89 agrees on discrete and compact groups with the ones we know provided it is infinite. In the case of finite rank we will have to inspect our classification of finite p-rank groups. The relevant aspect of any theory of a general p-rank is whether it allows us to define clearly whether it is finite or infinite and to determine it easily in the case it is finite. From Remark 3.90 we easily extract the following simple criterion which reduces the property of finite p-rank of a locally compact abelian p-group to the more familiar concepts of the p-rank of compact and of the p-rank of discrete (that is, abstract) abelian p-groups. Proposition 3.92. For a locally compact abelian p-group the following conditions are equivalent: (1) G has finite p-rank, (2) there is a compact open subgroup C such that both rankp (C) and rankp (G/C) are finite, (3) there is a compact open subgroup C such that C ≅ ℤm p ⊕ F for a nonnegative integer m ∈ ℕ0 and a finite abelian group F and the p-socle Sp (G/C) is isomorphic to ℤ(p)n for some n ∈ ℕ0 , (4) there is a natural number r such that every finitely generated subgroup of G can be generated by at most r elements. Lemma 3.93. Let G be a locally compact abelian torsion-free p-group containing an open subgroup U of finite p-rank. Then rankp (G) = rankp (U).

96 | 3 Abelian periodic groups Proof. Let r denote the p-rank of U. Then in light of Proposition 3.92 (4), for any finitely generated subgroup T we need to show that rankp (T) ≤ rankp (U). There is no loss of generality to assume T to contain U. As T, being finitely generated, is compact and contains U as an open subgroup, there is e ≥ 0 with pe T ⊆ U. The homomorphism sending x ∈ G to pe x is injective by the torsion-freeness assumption on G and therefore rankp (T) = rankp (pe T) ≤ rankp (U) = r follows.

3.11 Structure of locally compact p-groups of finite p-rank Lemma 3.94. Let K be a compact subgroup of a locally compact abelian p-group G such that G/K is divisible and torsion-free of finite p-rank. Then K splits algebraically and topologically, that is, G ≅ ℚnp ⊕ K. Proof. From Theorem 3.51 we have a compact torsion-free p-group C1 such that G/K ≅ ℚ⊗C1 . Therefore G contains a compact open subgroup containing K such that C/K = C1 and that G/C is discrete and divisible. Now Theorem 3.72 gives a splitting G = K1 ⊕ div(G). ̂ and the annihilator K ⊥ of K in G. ̂ By the anWe consider the character group G ⊥ nihilator mechanism (see [54], Theorem 7.64), K is the character group of G/K and ̂ ⊥ therefore is ≅ ℚm p for a nonnegative integer m. The factor group G/K is the character ̂ As a group of K and thus is discrete since K is compact. Now K ⊥ is open closed in G. ∗ ⊥ ∗ ̂ ̂ divisible subgroup of G it has a complement K and so G = K ⊕ K , algebraically. ̂ onto K ∗ has the open subgroup K ⊥ as kernel and so is continuThe projection of G ous. So K ⊥ splits algebraically and topologically. Hence K splits in G in the category of topological groups. In proceeding to the classification theorem piecemeal, let us say that a locally compact abelian group A is almost divisible if div(A) is closed and A/ div(A) is compact. Lemma 3.95. Let G be a locally compact abelian p-group satisfying the following assumptions: (i) the torsion group tor(G) is almost divisible and div(tor(G)) is discrete, (ii) the torsion-free quotient T = G/ tor(G) is almost divisible.

3.11 Structure of locally compact p-groups of finite p-rank | 97

Then there are sets I, J, and K, and there is a compact torsion group C such that G ≅ (ℚ ⊗ ℤIp ) ⊕ ℤJp ⊕ ℤ(p∞ )

(K)

⊕ C.

Proof. By hypothesis (i) and Lemma 3.13, infer that the subgroup div(tor G) is a direct summand of G, algebraically and topologically, which is isomorphic to ℤ(p∞ )(J) . For the remainder of the proof, we may and will assume that div(tor(G)) = {0} and that tor(G) is compact. By hypothesis (ii) and by Theorem 3.67 we have T = R ⊕ div(T) for a profinite torsion-free p-group R, and this latter subgroup therefore is ≅ ℤJp for some set J. By Theorem 3.51, div(T) ≅ ℚ ⊗ ℤIp for a set I. Let f : G → T be the quotient morphism modulo tor(G) and set H = f −1 (R). Then R = H/ tor(G) for some compact subgroup H of G, and G/H ≅ div(T) ≅ ℚ ⊗ ℤIp . Then I Lemma 3.94 implies G ≅ ℚm p ⊕ H. Since R ≅ ℤp is projective in the category of compact I p-groups, we have H ≅ ℤIp ⊕ tor(G) and G ≅ ℚm p ⊕ ℤp ⊕ tor(G), which proves the lemma. We recall the structure of the compact torsion subgroup C of G as a group Cp in formula (2) of Proposition 3.11 (ii) whose p-rank is rankp (C/pC). It is a good exercise to calculate the p-rank of the group G of Lemma 3.95. Let N = N1 ⊕ N2 ⊕ N3 ⊕ N4 be a sufficiently small compact open subgroup decomposed according to the direct sum I decomposition of Lemma 3.95. Then N1 ≅ ℤm p , N2 ≅ ℤp , N3 = {0}; and C/N4 is finite. Accordingly, the compact p-group N has the rank rankp (N) = m + card I + rankp (N4 /pN4 ). Correspondingly, the discrete abelian p-group G/N is isomorphic to m

(J)

ℤ(p∞ ) ⊕ F2 ⊕ ℤ(p∞ )

⊕ F4

for some finite p groups F2 and F4 and thus has the rank rankp G/N = m + rankp (F2 ) + card J + rankp (F4 ). So, if one of the cardinals I, J, rankp C is infinite, then rankp G = max{card I, card J, rankp C}. If, on the other hand, they are all finite, then for all sufficiently small N we have rankp N ≤ rankp G/N, S(F2 ) = ℤIp /pℤIp = card I and F4 = C so that rankp G = m + card I + card J + rankp C. As a consequence we have the following.

98 | 3 Abelian periodic groups Lemma 3.96. The p-rank of the group G of Lemma 3.95 is rankp G = m + card I + card J + rankp (C). Now we formulate the classification theorem for p-groups of finite p-rank due to V. S. Čarin; see Theorem 5 of [16]. Theorem 3.97 (Structure theorem on locally compact p-groups of finite rank). A locally compact abelian p-group G is of finite p-rank iff it is isomorphic to a group of the form n

ℚnp1 ⊕ ℤ(p∞ ) 2 ⊕ ℤnp3 ⊕ F for some nonnegative integers n1 , n2 , n3 and a finite p-group F of p-rank n4 . The p-rank of G is precisely rankp G = n1 + n2 + n3 + n4 . Proof. Let C be a compact open subgroup of G. Then rankp (C) < ∞. We observe tor C = C ∩ tor G. Then the socle S(tor(C)) is finite and since the compact p-group C cannot contain a Prüfer group, tor(C) is finite and, in particular, discrete. Since C is open and C ∩ tor G is finite, tor(G) is discrete. Since rankp (G/C) is finite and G/C is a discrete p-group, tor(G) is a discrete direct sum of a finite abelian group and a finite sum of Prüfer groups. Hence tor(G) satisfies hypothesis (i) of Lemma 3.95. Further, the torsion-free factor group T := G/ tor(G) is of finite rank and thus satisfies hypothesis (ii) of Lemma 3.95. So Lemma 3.95 applies with I and J finite sets and yields finite sets I and J, a nonnegative integer m, and a finite group F such that ∞ I G = ℚm p ⊕ ℤp ⊕ ℤ(p )

(J)

⊕ F.

This proves the first part of the theorem with the obvious choices for the quadruple n1 , n2 , n3 , n4 . Finally, Lemma 3.96 establishes the claim on the p-rank of G. Every finitely generated locally compact abelian p-group has finite rank. Since neither ℚkp nor ℤ(p∞ )m is finitely generated, from Theorem 3.97 we recover at once the observation that a finitely generated locally compact abelian p-group is of the form ℤnp ⊕ F for a natural number n and a finite p-group: See Proposition 3.9. Since a periodic locally compact abelian group is a local product of its Sylow groups, we obtain at once the following. Corollary 3.98. A finitely generated locally compact abelian periodic group G is of the form loc

G = ∏ (Gp , Cp ) p prime

where the p-ranks of the p-Sylow subgroups Gp are bounded by some natural number.

3.11 Structure of locally compact p-groups of finite p-rank | 99

The next consequence will be needed in Section 14.4.1. Lemma 3.99. Let G be a reduced locally compact abelian p-group containing an open compact finitely generated subgroup U. Then the torsion subgroup tor(G) is discrete and hence closed and G is algebraically and topologically a direct sum G = ℤkp ⊕ L for some nonnegative integer k and L containing tor(G) with factor group L/ tor(G) divisible and torsion-free of finite p-rank. Proof. Since U is compact and finitely generated its torsion subgroup is finite. Hence, replacing U by a smaller open subgroup we can arrange that tor(G)∩U = {0}. Therefore tor(G) is a discrete and hence closed subgroup of G. Then the torsion-free factor group G/ tor(G) contains the compact open subgroup (U +tor(G))/ tor(G) of finite p-rank. As a consequence of Lemma 3.93 the p-rank of G/ tor(G) turns out to be finite. Theorem 3.97 in conjunction with Proposition 1.27 implies that we can find an algebraic and topological direct decomposition G = K ⊕ L, where K is a finitely generated torsion-free compact p-group and L contains tor(G) with L/ tor(G) torsion-free and divisible of finite p-rank. For the second summand the factor group L/ tor(G) can well be nontrivial as becomes clear from Theorem 3.17 (iv)+(viii). Question 3.100. Let us conclude by asking for natural and systematic conditions that would secure when, in a locally compact abelian p-group G, such characteristic subgroups as, e. g., the torsion subgroup tor(G) or the largest divisible subgroup div(G) of G are closed subgroups and when they are algebraic and topological direct summands of G.

4 Scalar automorphisms and the mastergraph If G is a periodic locally compact abelian group, then every automorphism (and indeed every endomorphism) α leaves the Sylow subgroup Gp invariant. We write αp = α|Gp : Gp → Gp . If C is a compact open subgroup, let End(G, C) denote the subring of the ring End(G) of all endomorphisms leaving C invariant. In view of Theorem 3.3 we may identify G with its canonical local product decomposition of the pair (G, C). Proposition 4.1. For a periodic locally compact abelian group G, the componentwise application κ defined by α 󳨃→ (αp )p : End(G, C) → ∏ End(Gp , Cp ) p

is an isomorphism of groups, and α((gp )p ) = (αp (gp ))p . Proof. After identifying (G, C) and ∏loc p (Gp , Cp ) according to Theorem 3.3, it is straightforward to verify that κ is an injective morphism of groups. Moreover, if (ϕp )p ∈ ∏ End(Gp , Cp ), p

then the morphism ϕ: ∏ Gp → ∏ Gp p

p

defined by ϕ((gp )p ) = (ϕp (gp ))p

leaves C = ∏p Cp fixed as a whole and does the same with ∏loc p (Gp , Cp ) and so κ(ϕ) = (ϕp )p . Thus κ is surjective as well. Remark 4.2. The Sylow theory for profinite groups implies a unique product representation ̃ = ∏ ℤp . ℤ p∈π

Every locally compact abelian p-group A is a ℤp -module for a multiplication ̃ and ∏ (rp , gp ) 󳨃→ rp ⋅gp . If we identify ℤ p∈π ℤp by (4.2) and a periodic locally compact

abelian group A with ∏loc p∈π (Ap , Cp ) for any compact open subgroup C, we see at once that we have a continuous module multiplication ̃ × A → A. (r, g) = ((rp )p , (gp )p ) → (rp ⋅gp )p = r⋅g : ℤ Recall the meaning of νp (n) for n ∈ ℕ and p a prime from Notation 3.1. https://doi.org/10.1515/9783110599190-004

(4.1)

102 | 4 Scalar automorphisms and the mastergraph Lemma 4.3. Let p ∈ π be an arbitrary prime number and n an arbitrary natural number. Then there is a prime number q such that n ≤ νp (q−1). Accordingly, pn |(q−1). In particular p|(q − 1). Proof. Fix p ∈ π and an arbitrary natural number n. The numbers a = pn and b = 1 are relatively prime. Hence the arithmetic progression (am + b)m∈ℕ contains infinitely many primes q by the Dirichlet prime number theorem. Let q be one of them. Then q − 1 = pn m, that is, νp (q − 1) ≥ n. In particular, p|(q − 1). ̃ × contains for each fixed prime p a Lemmas 4.13 and 4.16 imply via (4.11) that ℤ product E := ∏ ℤ(pνp (q−1) ), q∈π

where we note that νp (q − 1) = 0 if p fails to divide q − 1. Therefore the following conclusion of the preceding Lemma 4.3 is relevant. Proposition 4.4. Let p ∈ π be an arbitrary prime number. Then the group E contains a subgroup isomorphic to P = ℤ(p) × ℤ(p2 ) × ℤ(p3 ) × ⋅ ⋅ ⋅ which in turn contains a dense subgroup and ℤp -module D ≅ ℤ(ℕ) p . Proof. By Lemma 4.3 for each n there is a q ∈ π such that n ≤ νp (q −1). Hence the group ℤ(pνp (q−1) ) contains a subgroup Bn ≅ ℤ(pn ). Thus E contains an isomorphic copy of B = ∏ Bn ≅ ℤ(p) × ℤ(p2 ) × ℤ(p3 ) × ⋅ ⋅ ⋅ . n∈ℕ

The remainder then follows from Corollary 3.20. ̃ yields an endomorphism a 󳨃→ r⋅a of the periodic We noted in (4.1) that every r ∈ ℤ ̃ → End(A). In locally compact abelian group A, giving us a morphism of rings ζ : ℤ ̃ × A → A is continuous, ker(ζ ) is a closed particular, since scalar multiplication ℤ ̃ ideal of ℤ. Definition 4.5. For a locally compact abelian group A we denote the factor ring ̃ ker(ζ ) by ℛ(A) and call it the ring of scalars of A. There is an obvious scalar multiℤ/ plication ℛ(A) × A → A. The ring morphism ζ factors through an isomorphism of rings ≅

ℛ(A) 󳨀 󳨀󳨀󳨀→ End(A).

(4.2)

ℛ(A) ≅ ∏ ℛ(A)p ,

(4.3)

We note that p

and scalar multiplication operates componentwise on A ≅ ∏p (Ap , Cp ).

4.1 On scalar automorphisms | 103

̃ In this We shall be mostly interested in the scalar multiplication by units r ∈ ℤ. context it is clear that (4.3) induces an isomorphism ×

×

ℛ(A) ≅ ∏(ℛ(A)p ) , p

(4.4)

where (ℛ(A)p )× is isomorphic to a quotient group of (ℤp )× . One verifies easily the following piece of information. Example 4.6. Let A be a locally compact abelian p-group. Then ℤp /pm ℤp ≅ ℤ(pm ), if A has finite exponent pm ,

ℛ(A) = {

otherwise.

ℤp ,

(4.5)

4.1 On scalar automorphisms We record some essential definitions which we have to keep in mind throughout. The following observation is elementary. Lemma 4.7. For a continuous automorphism α of a locally compact group G the following conditions are equivalent: (1) α(H) ⊆ H for all closed subgroups H of G, (2) α(⟨g⟩) ⊆ ⟨g⟩ for all g ∈ G, (3) α(g) ∈ ⟨g⟩ for all g ∈ G. Definition 4.8. An automorphism α of a locally compact group G is called scalar if it satisfies the equivalent conditions of Lemma 4.7. The subgroup of scalar automorphisms of G is denoted SAut(G). If G is abelian and is written additively, then the subgroup {idG , −idG } ⊆ SAut(G) is said to consist of trivial scalar automorphisms. All other scalar automorphisms are called nontrivial. Proposition 4.9. For a locally compact group G with a periodic abelian closed normal subgroup A the following statements are equivalent: (i) Int(A) ⊆ SAut(A), (ii) every inner automorphism induced on A is a scalar automorphism, (iii) every closed subgroup of A is normal in G, (iv) there is a morphism ρ: G → ℛ(A)× such that (∀g ∈ G, a ∈ A) gag −1 = ρ(g)⋅a.

104 | 4 Scalar automorphisms and the mastergraph For a proof of this proposition see Lemma 4.7. We emphasize here again that in Theorem 4.43 we shall give an explicit structure theory of ℛ(A)× ≅ SAut(A). Notice that we shall not only call the identity automorphism, but also the inversion automorphism “−idG ” trivial. Every locally compact abelian group has these trivial scalar automorphisms, but it can happen that −idG = idG . Indeed this occurs if and only if every element of G has order 2 (for the structure of these groups see Section 8.1, notably Corollary 3.26 (a)). More generally we record a few remarks pertinent to the trivial scalar automorphism −idA . For later use we formulate them a bit more generally. Remark 4.10. For the action of the involution τ = −idA on a locally compact abelian group A, two characteristic subgroups arise. Firstly, the group of fixed points of τ is the kernel of the endomorphism 2: A → A, 2(a) = 2⋅a. This kernel is exactly the 2-socle S2 (A) = {a ∈ A : 2⋅a = 0} of A (cf. [54], Definition A1.20). Secondly, the closed subgroup generated by {a − τ(a) : a ∈ A} is precisely the closure of the image of 2. The example of the local power ∏loc ℕ (ℚ2 , ℤ2 ) shows that im(2) may be nonclosed and dense in A. In general, the subgroup 2−1 (S2 (A)) is exactly the subgroup of all elements of order 4. Braconnier’s Theorem 3.3 allows us to identify the issue of im(2) in the case of a periodic locally compact abelian group A. Such an A we can decompose into a direct product A = A2 × B2 where B2 := ∏loc p>2 (Ap , Cp ) for some compact open subgroup Cp of the p-primary component Ap of A. For each prime p > 2 the endomorphism 2 induces an automorphism on Ap , since the scalar 2 ∈ ℤp is a unit. Thus S2 (A) = S2 (A2 × {0}) and im(2)A = im(2A2 ) × B2 . In general, by duality, for any locally compact abelian group G, the isomorphism ̂ maps SAut(G) bijectively onto SAut(G). ̂ The set SAut(G) is ̂ : Aut(G) → Aut(G) α 󳨃→ α a closed subgroup of Aut(G) where the latter is equipped with the coarsest topology containing the compact open topology and turning Aut(G) into a topological group (the “g-topology” in [70]). If C is any compact open subgroup of G, then SAut(G) ⊆ Aut(G, C). Corollary 4.11. For a periodic locally compact abelian group G and any compact open subgroup C, with the identification loc

G = ∏(Gp , Cp ), p

the function α 󳨃→ (αp )p : SAut(G) → ∏ SAut(Gp ) p

is an isomorphism of groups.

4.1 On scalar automorphisms | 105

Proof. Since every closed subgroup H of G satisfies Hp = H ∩ Gp and H = ∏loc p (Hp , Hp ∩ C), the assertion is a consequence of Corollary 4.1. Thus we know SAut(G) if we know SAut(Gp ) for each prime p. Recall from Definition 2.2 that a locally compact group G is called a p-group if each of its elements is a p-element, that is, generates a compact p-group ⟨g⟩. Each locally compact abelian p-group G is a ℤp -module and is compactly ruled. By Corollary 1.12, every solvable locally compact p-group is compactly ruled. For any invertible scalar r ∈ ℤ×p the function μr : G → G, given by μr (g) = r⋅g, is a scalar automorphism. Lemma 4.12. For a locally compact abelian p-group G the morphism r 󳨃→ μr : ℤ×p → SAut(G) is surjective for all prime numbers p including p = 2. Proof. Let α ∈ SAut(G). We must find an r ∈ ℤ×p such that α = μr . If G is compact, this follows from Theorem 2.27 of [57]. Assume now that G is not compact. Then it is the union of its compact subgroups. If C is a nonsingleton compact subgroup, then by Theorem 2.27 of [57] there is a unique scalar rC ∈ ℤ×p such that α|C = μrC |C. If K ⊇ C is a compact subgroup containing C, then μrK |C = μrC |C. Since ℤ×p is compact, the net (rC )C , as C ranges through the directed set of compact subgroups of G, has a cluster point r ∈ ℤ×p ; that is, there is a cofinal subset C(j), j ∈ J, for some directed set J such that r = limj rC(j) . Since for every g ∈ G there is a C with g ∈ C and then a j0 ∈ J such that j ≥ j0 implies C ⊆ C(j) we conclude that α(g) = rC ⋅g = rC(j) ⋅g for these j, and so α(g) = r⋅g. Thus α = μr and the proof is complete. There is an isomorphism σ : SAut(G) → ℛ(G)× . For a compact abelian p-group A, the procyclic group SAut(A) of scalar automorphisms is amply discussed in [57]; it amounts to a precise record of the group of units of the ring ℛ(A). We next discuss this for the p-adic rings of integers ℤp , first for odd p and later for p = 2.

4.1.1 The structure of ℤp for p ≠ 2 For a unital commutative ring R we denote by R× the multiplicative group of its units, i. e., invertible elements. We clarify this concept for R = ℤp by a reminder of some elementary structural information of ℤp . Recall that under suitable circumstances in a topological ring R the sequence 1 + x + 21 ⋅x + 3!1 x3 ⋅ ⋅ ⋅ converges for x from a suitable domain D and defines a function exp: D → 1 + D,

1 + D ⊆ R× .

106 | 4 Scalar automorphisms and the mastergraph If p ∈ π is a prime and m ∈ ℕ, then recall from Notation 3.1 νp (m) = max{n ∈ ℕ0 : pn |m}

(4.6)

is that unique nonnegative integer n for which m = pn m󸀠 and (m󸀠 , p) = 1. Background information on ℤp will be used all the way. For easy reference we recall some of it here. The character group of ℤp is the Prüfer group ℤ(p∞ ) = p1∞ ℤ/ℤ ⊆ 𝕋 := ℝ/ℤ. The proper nonzero subgroups of the Prüfer group are the cyclic groups 1/pm ℤ/ℤ ≅ ℤ(pm ), m ∈ ℕ, and, accordingly, the closed nonzero proper additive subgroups of ℤp are the open subgroups pm ℤp , m ∈ ℕ. All nontrivial factor groups of the Prüfer group are isomorphic to itself; for each m ∈ ℕ, the function z 󳨃→ pm z: ℤp → pm ℤp is an isomorphism of compact abelian groups. We need the structure theory of the multiplicative group of units ℤ×p of the ring ℤp of p-adic integers and the easy passage between the additive and multiplicative structure of ℤp as we shall discuss presently. (See [32], Chapter 8 and [9], A.VII.12 § 2, No. 4.) The multiplicative subgroup (1 + pℤp , ×) of ℤ×p will play an important role. We shall use the abbreviation ℙp := (1 + pℤp , ×)

for all primes p.

Any automorphism of ℤp will leave ℙp invariant, that is, ℙp is a characteristic subgroup of the multiplicative group ℤ×p of units of the ring ℤp . For p ≠ 2, the group ℙp is the p-primary component of ℤ×p . For the following lemma see, e. g., [9], p. A.VII.14 in a slightly different notation. The additive subgroup ⟨Pp ⟩ is open and contains 1, hence agrees with ⟨1⟩ = ℤp . For the following information on the ring ℤp of p-adic integers see, e. g., [57]. Lemma 4.13. (i) For each prime p ≠ 2, the function exp: p⋅ℤ → (1 + pℤp ),

1 + pℤp ⊆ ℤ×p ,

is an isomorphism of profinite groups and 1 + pℤp is an open subgroup of ℤ×p . In particular, z 󳨃→ exp pz : (ℤp , +) → (1 + pℤp , ×)

(4.7)

is an isomorphism of profinite groups. (ii) The factor ring ℤp /pℤp is the field GF(p) of p elements, and so (ℤp /pℤp )× is a cyclic group of p − 1 elements. The ring ℤ×p contains a cyclic group Cp of p − 1 elements (called roots of unity) such that (x, c) 󳨃→ xc : (1 + pℤp ) × Cp → ℤ×p

is an isomorphism,

4.1 On scalar automorphisms | 107

and (ℤ×p , ×) ≅ (ℤp ⊕ ⨁ ℤ(qνq (p−1) ), +).

(4.8)

q∈π

In particular, for q ∈ π the q-Sylow subgroup of ℤ×p is procyclic and ℤ(qνq (p−1) ), { { { (ℤ×p )q ≅ {ℤp , { { {{0},

if q < p, if q = p, if p < q.

Lemma 4.14. For p ≠ 2 the group ℤ×p of units of ℤp is the direct product ℙp × Cp−1 of the subgroup ℙp and a finite cyclic group Cp−1 which is the cyclic group of (p − 1)th roots of unity. Lemma 4.15. The function exp : (pℤp , +) → ℙp ,

1 exp pz = 1 + pz + (pz)2 + ⋅ ⋅ ⋅ 2

is an isomorphism of compact abelian groups and has the inverse

1 1 1 − pz 󳨃→ log(1 − pz) = pz + (pz)2 + (pz)3 + ⋅ ⋅ ⋅ . 2 3

In particular, ℙp is isomorphic to the additive group of ℤp . Therefore ℙp is the p-component of ℤ×p . Regarding the definition of the exponential function and the logarithm in the domain of p-adic numbers see [57, p. 651–653]. The isomorphism exp: pℤp → ℙp maps the subgroups pm ℤp isomorphically onto the subgroups m ℙ[m] p := (1 + p ℤp , ×),

m = 1, 2, 3, . . . .

Since z 󳨃→ pz : ℤp → pℤp is an isomorphism, all subgroups in sight are isomorphic to ℤp ; hence ℙ[m] p ≅ ℙp ≅ ℤp

for all primes p.

The group ℤ×p of units of ℤp is naturally isomorphic to the group of automorphisms of ℤp by acting under scalar multiplication. Recall that ℤp /pℤp is the field GF(p) of p elements. There are two exact sequences 0 1

pℤp ↑ ↑ exp↑ ↑ ↓ → ℙ

→

incl

quot

󳨀󳨀󳨀󳨀→

ℤp

󳨀󳨀󳨀󳨀→

GF(p)

→

0,

󳨀󳨀󳨀󳨀→

ℤ×p

󳨀󳨀󳨀󳨀→

Cp−1

→

1.

incl

The top sequence reflects the ring level, whereas the bottom sequence is a split exact sequence of multiplicative groups of units.

108 | 4 Scalar automorphisms and the mastergraph 4.1.2 The structure of ℤ2 While the additive group of ℤ2 is analogous to ℤp for p > 2, the multiplicative group 1 n of units ℤ×2 is different. One of the reasons is that the infinite series ∑∞ n=0 n! (2z) fails × to converge for z ∈ ℤ2 \ 2ℤp = ℤ2 = (1 + 2ℤ2 , ×) in general. So we do not have an exponential function from 2ℤ2 to ℤ×2 . However, if we set ℙ4 = (1 + 4ℤ2 , ×) ⊆ ℤ×2 and C2 = ({1, −1}, ×), we can formulate the following description. Lemma 4.16. (i) The function exp: 4ℤ2 → (1 + 4ℤ2 ),

1 + 4ℤ2 ⊆ ℤ×2

is an isomorphism of profinite groups and 1 + 4ℤ2 is an open subgroup of ℤ×2 . In particular, z 󳨃→ exp 4z : (ℤ2 , +) → (1 + 4ℤ2 , ×)

(4.9)

is an isomorphism of profinite groups. (ii) The factor ring ℤ2 /2ℤ2 is the field GF(2) of 2-elements, and the group of units of (ℤ2 /4ℤ2 )× is a group of 2 elements. The group ℤ×2 contains a cyclic group C2 of 2-elements (called roots of unity) such that (x, c) 󳨃→ xc : (1 + 4ℤ2 ) × C2 → ℤ×2

is an isomorphism,

and (4.10)

(ℤ×2 , ×) ≅ (ℤ2 ⊕ ℤ(2), +). In particular, ℤ×2 is a nonprocyclic 2-group. The product representation in Remark 4.2 immediately yields ̃ × = ∏ ℤ× . ℤ p

(4.11)

p∈π

Since for p ≠ 2 the profinite group ℤ×p is not a p-group, the product representation ̃ × in (4.11) is not its Sylow decomposition. Our first and foreof the profinite group ℤ ̃ × and to describe it in an most goal is now to determine the Sylow decomposition of ℤ intuitive and useful form. There are two exact sequences (the bottom sequence splits, the top sequence does not): 0

2ℤ2 ↑ ↑ exp 2(−) ↑ ↑ ↓ 1 → ℙ4 →

incl

quot

󳨀󳨀󳨀󳨀→

ℤ2

󳨀󳨀󳨀󳨀→

󳨀󳨀󳨀󳨀→

ℤ×2

󳨀󳨀󳨀󳨀→

incl

pr2

ℤ(2) ↑ ↑ ↑ ↑≅ ↓ C2

→

1,

→

1.

4.1 On scalar automorphisms | 109

The remarkable difference in the structures of the p-components of the multiplicative groups ℤ×p between the cases p = 2 and p > 2 will compel us to deal with the case p = 2 separately. With this terminology we have the following. Proposition 4.17. For a compact p-group A, via the natural morphism σA : ℤ×p → SAut(A), the procyclic group SAut(A) is isomorphic to ℤ×p ≅ ℙp × Cp−1 , { { { { { {ℤ×2 ≅ ℙ4 × C2 , { { ℤ(pn )× ≅ (ℙp /ℙ[n] { p ) × Cp−1 , { { { [n] n × {ℤ(2 ) ≅ (ℙ4 /ℙ4 ) × C2 ,

if σA is faithful, p > 2,

if σA is faithful, p = 2,

if σA is not faithful, p > 2, n > 1,

if σA is not faithful, p = 2, n > 2.

Remark 4.18. We use the opportunity to correct a typographical error in the proof of [57], Theorem 2.27. In that presentation, SAut(A) is called Autscal (A) and the formula for a description of this group in terms of p-adic units should read as above in Proposition 4.17. The validity of the arguments in [57] is not affected by the typographical flaw we correct here. According to Lemma 4.15, for p > 2 and n > 1, we have ℙp ℙ[n] p

≅

pℤp

pn ℤp

≅ ℤ(pn−1 ),

and according to Lemma 4.16, for p = 2, n > 2, similarly, ℙ4 ℙ[n] 4

≅

4ℤ2 ≅ ℤ(2n−2 ). 2n ℤ2

Accordingly, we record the following corollary of Proposition 4.17. Corollary 4.19. The group of units of the finite ring ℤ(pn ), in additive notation, is as follows: For p > 2, n ∈ ℕ, ℤ(pn ) ≅ ℤ(pn−1 ) × ℤ(p − 1), ×

and for {0}, for n = 1, × ℤ(2n ) ≅ { n−2 ℤ(2 ) × ℤ(2), n ≥ 2. The group ℤ(2n ) is a 2-group, while for p > 2 the p-primary component (i. e., p-Sylow subgroup) is ℤ(pn−1 ).

110 | 4 Scalar automorphisms and the mastergraph Corollary 4.20. Let p be any odd prime. Then the group of units of ℤp and of any of its homomorphic images ℤ(pn ) is monothetic. ̃ we The universal zero-dimensional compactification of the integers is denoted ℤ; ̃ ≅ ∏ ℤp , where p ranges through the set of all prime numbers. When, know that ℤ p here, as before, and in the following, in expressions like ∏ ℤp , p

⨁ ℤ(p∞ ), p

(xp )p ∈ ∏ ℤp , p

it will be understood that p ranges through the set of all primes whenever no confusion arises. We record the following facts on periodic locally compact groups which we derive from Remark 2.1 and which again we shall utilize in the abelian case. Proposition 4.21. Every element g in a periodic locally compact group generates a procyclic group ⟨g⟩ ≅ ∏p ⟨g⟩p and thus may be naturally identified with (gp )p ∈ ∏p ⟨g⟩p . ̃ ̃= This “decomposition” gives rise to a ℤ-action on G by our considering z = (zp )p ∈ ℤ ∏p ℤp and defining z⋅g by (∀p ∈ π(G)) (z⋅g)p = zp ⋅gp . ̃ This turns G into a ℤ-space with a continuous linear action ̃→G z 󳨃→ z⋅g : ℤ

for all g ∈ G.

̃ × G → G is bilinear and continuous. If G is abelian, the action (z, g) 󳨃→ z⋅g : ℤ From Proposition 4.21 we note what we had observed in [57] for the special case of compact p-groups. Proposition 4.22. Let G a periodic locally compact group G. For each g ∈ G we have ̃ ⟨g⟩ = ℤ⋅g.

4.2 The structure of the group of scalar automorphisms In the next proposition we show that for a locally compact abelian periodic group G ̃ × , and we investigate the case when all the group SAut(G) is a quotient group of ℤ scalar automorphisms are trivial: Information about this situation will be important. Proposition 4.23. Let the locally compact abelian group G be periodic. Then we have the following conclusions: ̃ × → SAut(G) is surjective. In particular, SAut(G) is a profinite (i) The natural map ζ : ℤ ̃×. group and a homomorphic image of ℤ

4.2 The structure of the group of scalar automorphisms | 111

(ii) The subsequent two statements are equivalent: (a) SAut(G) = {idG , −idG }, (b) The exponent of G is 2, 3, or 4. Notably: The exponent of G is 2 if and only if −idG = idG . Proof. (i) This follows from Corollary 4.11 and Lemma 4.12. (ii) If the exponent of G divides 4 or is 3, then any α ∈ SAut(G) satisfies (∀x ∈ G) α(x) = x or (∀x ∈ G) α(x) = −x. So such an α is “trivial” in the sense of Definition 4.8. On the other hand, if the exponent is not in {2, 3, 4}, then at least one of the following statements holds: (α) There is a p > 3 such that Gp ≠ {0}, (β) G3 has exponent at least 9, (γ) G2 has exponent at least 8. In case (α), Cp−1 has order p − 1 ≥ 4 and acts nontrivially on Gp as group of scalar automorphisms of G. In case (β), the map x 󳨃→ 4x is a nontrivial scalar automorphism of G3 , and in case (c), the function x 󳨃→ 5x is a nontrivial scalar automorphism of G2 . Thus, since a nontrivial automorphism of any primary component Gq gives a nontrivial automorphism of G, if the exponent of G is not in {2, 3, 4}, then there are nontrivial scalar automorphisms. We note that for p = 2 the group SAut(G2 ) is not monothetic if the exponent of G is at least 8 (see Proposition 4.17). A proof of part (i) of Proposition 4.23 can also be obtained as a corollary from a more general result by R. Winkler [114]. Every (not necessarily continuous) endomorphism of a periodic locally compact abelian group which leaves invariant every closed subgroup must act by scalars. In the same line of structural results we point out a note by Moskalenko [76], which has an echo in the following sharper result. Theorem 4.24. For a locally compact abelian group G, we consider the following statements: (1) G has nontrivial scalar automorphisms, (2) G is periodic. Then (1) implies (2), and if G does not have exponent 2, 3, or 4, then both statements are equivalent. Proof. We remarked in Proposition 4.23 that (2) implies (1) under the hypotheses stated there. Now assume (1). If G has a discrete subgroup ≅ ℤ, then every scalar automorphism must be trivial. Hence Weil’s lemma (see, e. g., [54], Proposition 7.43) implies that G = ̂ is comp(G) ̂ (see [54], comp(G) and we have to argue that G0 = {0}. But (G0 )⊥ in G

112 | 4 Scalar automorphisms and the mastergraph ̂ has no nontrivial scalar automorphisms by (1), we know G ̂ = Theorem 7.67). Since G ⊥ ̂ = (G ) . This means G = {0}, as we had to show. comp(G) 0

0

Theorem 4.25 is a quite detailed version of [79, Theorem 2], but may now also be obtained quickly by combining our results in Proposition 4.17, Corollary 4.19 with the results in the present Section 4.2. For the ring ℛ(G) of scalars of a locally compact p-group see Definition 4.5. Theorem 4.25 (Mukhin, Theorem 2 in [79]). Let G be a locally compact abelian group written additively. Then ̃×, (a) SAut(G) is a homomorphic image of ℤ (b) if G is not periodic, then SAut(G) = {id, −id}, (c) if G is periodic, then SAut(G)= ∏p SAut(Gp ), where SAut(Gp ) may be identified with the group of units of the ring ℛ(Gp ) of scalars of Gp , namely, ℤp × ℤ(p − 1), { { { { m−1 { { {ℤ(p ) × ℤ(p − 1), { { × ℛ(Gp ) ≅ {ℤ2 × ℤ(2), { { { { ℤ(2m−2 ) × ℤ(2), { { { { {{0},

if p > 2 and the exponent of Gp is infinite,

if p > 2 and the exponent of Gp is pm ,

if p = 2 and the exponent of G2 is infinite,

if p = 2 and the exponent of G2 is 2m > 2,

if p = 2 and the exponent of G2 is 2,

̃ × such that (d) an α ∈ Aut(G) is in SAut(G) iff there is a unit z ∈ ℤ (∀g ∈ G) α(g) = z⋅g = ∏ zp ⋅gp p

for z = ∏ zp , g = ∏ gp . p

p

From part (c) it is again clear that the decompositions ̃ × = ∏ ℤ× ℤ p p

and

SAut(G) = ∏ SAut(Gp ) p

̃ respectively, SAut(G). are not the prime decompositions of ℤ, Some warnings may be called for. Remark 4.26. In order to avoid confusion let us note that the “usual scalar multiplication” in ℝ does not yield a “scalar automorphism” of ℝ in the present sense: If 0 ≠ r is any real number, then multiplication by r is a scalar automorphism in our present sense if and only if r = ±1. The notion of a “scalar automorphism” as defined in Definition 4.8 may lead to another misconception suggested by scalar automorphisms on vector spaces, namely, that a scalar automorphism τ ∈ SAut(A) on a locally compact abelian group A (which we write additively) could not have nonzero fixed points as soon as τ ≠ idA . Recall Notation 2.34 in the sequel.

4.2 The structure of the group of scalar automorphisms | 113

Remark 4.27. For A any locally compact abelian group and every τ in SAut(A), α = τ − 1 is an endomorphism preserving all monothetic subgroups so that CA (τ) = ker α. According to Remark 4.10, if τ = −idA , then CA (τ) = S2 (A) is the 2-socle {a ∈ A : 2⋅a = 0} of A. Example 4.28. (a) If p and q are two prime numbers with p + 1 < q, then A = ℤ(pq) = q⋅A ⊕ p⋅A with p⋅A = Aq ≅ ℤ(q) and q⋅A = Ap ≅ ℤ(p). So τ: A → A, τ(x) = (p + 1)⋅x is a scalar automorphism of A since p + 1 is the identity modulo p and a unit modulo q. But Ap ≅ ℤ(p) is the kernel of α = τ − 1, i. e., CA (τ) = q⋅A. (b) On the cyclic group A = ℤ(4n) = ℤ(4) × ℤ(n), (2, n) = 1, the scalar automorphism −idA has as proper fixed point set CA (−idA ) = 2⋅ℤ(4) × {0} ≅ ℤ(2). (See Remark 4.10.) We now link the concepts of scalar automorphisms of a locally compact group and its Chabauty space. So let G be an arbitrary totally disconnected locally compact group. Then the function μ: G → 𝒮𝒰ℬ(G), μ(g) = ⟨g⟩, is continuous (see [58]). For x ∈ G we write μx (g) = x−1 ⟨g⟩x. Then the function Fx : G → G × 𝒮𝒰ℬ(G),

Fx (g) = (g, μx (g))

is likewise continuous. The binary relation of element containment ∈ {(g, A) : g ∈ A} ⊆ G × 𝒮𝒰ℬ(G) is a closed subset of G × 𝒮𝒰ℬ(G). Thus for x ∈ G the set Bx := {g ∈ G : xgx−1 ∈ ⟨g⟩} = {g ∈ G : g ∈ μx (g)} = Fx−1 (∈) is closed in G. Definition 4.29. We shall write B(G) := ⋂ Bx = {g ∈ G : (∀x ∈ G) xgx−1 ∈ ⟨g⟩} = {g ∈ G : ⟨g⟩ ◁ G}. x∈G

From the fact that all sets Bx are closed in G we find that B(G) is a closed subset of G. In view of Lemma 4.7 we have a useful reformulation of the action of the group of inner automorphisms on a subgroup by scalar action. Proposition 4.30. Let G be a locally compact group and A a closed subgroup. Then the following conditions are equivalent: (1) every closed subgroup of A is normal in G, (2) the group of inner automorphisms of G acts on A by scalar automorphisms, (3) A ⊆ B(G). Proof. The equivalence of (1) and (2) is clear from Definition 4.8 and Lemma 4.7; the latter and the preceding definitions show the equivalence of (2) and (3).

114 | 4 Scalar automorphisms and the mastergraph Corollary 4.31. Let G be a locally compact group. Then the set of all closed abelian subgroups A in which all closed subgroups are normal in G is a closed subset of 𝒮𝒰ℬ(G).

4.3 A bipartite graph for scalar action on a periodic locally compact abelian group An automorphic action of a locally compact abelian periodic group Γ on a locally compact abelian group A amounts to a morphism σ: Γ → SAut(A). We shall have to deal with those actions systematically. The group Γ has a primary decomposition ∏loc p (Γp , Kp ) and we know the morphism σ if we know its restriction to the primary components Γp . However, while we know that SAut(A) ≅ ∏p SAut(Ap ) according to Corollary 4.11, each factor SAut(Ap ) in this decomposition is a factor group of the procyclic group ℤ×p which itself has a relatively involved primary decomposition as we know from Proposition 4.17. Thus the p-primary decomposition of SAut(A) is combinatorially rather complex. This fact has not explicitly emerged in the previous discussion or, to the best of our knowledge, in past literature. ̃ × and The method we propose in order to deal with the Sylow structure of ℤ SAut(A) is graph theoretical and follows next. We introduce a bipartite edge labeled graph 𝒢 as follows. Definition 4.32. A bipartite graph consists of two disjoint sets U and V and a binary relation ℰ ⊆ (U ∪ V)2 such that (u, v) ∈ ℰ implies u ∈ U and v ∈ V. The elements of U ∪ V are called vertices and the elements of ℰ are called edges. Any triple (U, V, ℰ ) of this type is called a bipartite graph. An edge labeled graph is a quadruple (U, V, ℰ , λ) such that (U, V, ℰ ) is a bipartite graph and λ is a function λ: ℰ → L for some set L of labels. Labels could be numbers, or symbols like ∞. Now we define a particular edge labeled graph 𝒢 . Recall the definition of νp (m) from (4.6). Definition 4.33. The bipartite edge labeled graph 𝒢 = (U, V, ℰ , λ),

ℰ ⊆ U × V,

will be called the prime mastergraph or mastergraph for short. We have (i) U = π × {1} ⊆ π × {0, 1}, (ii) V = π × {0} ⊆ π × {0, 1}, (iii) ℰ = {((p, 1), (q, 0)) : p = q or p|(q − 1)}, (iv) λ: ℰ → ℕ ∪ {∞}, 0, if p = q, λ(((p, 1), (q, 0))) = { νp (q − 1), if p < q.

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| 115

We shall call the vertices in U the upper and those in V the lower vertices. The edges ((p, 1), (p, 0)), p ∈ π are said to be vertical, all others are called sloping. We say that e = ((p, 1), (q, 0)) is the edge from p to q.

4.3.1 Geometric properties of the mastergraph The “geometric” terminology is chosen because 𝒢 has an intuitive representation in the plane ℝ2 preserving the order. Proposition 4.34. Let ω: π → ℕ be the bijection inverse to the usual enumeration n 󳨃→ pn of primes according to their natural ordering according to their size. Let id be the identity map of the set {0, 1}. There is a faithful representation of the configuration of 𝒢 into the plane ℝ2 preserving the componentwise order which is induced by the injection ω×id

incl

π × {0, 1} 󳨀󳨀󳨀󳨀→ ℕ × {0, 1} 󳨀󳨀󳨀󳨀→ ℝ × ℝ = ℝ2 and taking U to ℕ × {1} and V to ℕ × {0}. In Figure 4.1 the label of the edge from (2, 1) to (13, 0) is 2. Definition 4.35. Let p and q be any primes. Then ℰp = {e : e = ((p, 1), (p , 0)) ∈ ℰ such that p = p or p|(p − 1)}, 󸀠

󸀠

󸀠

the set of all edges emanating downwards from the vertex (p, 1) ∈ U will be called the cone peaking at p. Further the set ℱq = {e : e = ((q , 1), (q, 0)) ∈ ℰ such that q |(q − 1)}, 󸀠

󸀠

the set of edges ending below in the vertex (q, 0) ∈ V is called the funnel pointing to q. 2

3

5

7

11

13

2

3

5

7

11

13

Figure 4.1: The initial part of the mastergraph.

116 | 4 Scalar automorphisms and the mastergraph

2

3

5

7

2

3

5

7

2

3

5

7

2

3

5

7

Figure 4.2: Vertical and sloping edges.

2

3

5

7

⋅⋅⋅

211

2

3

5

7

⋅⋅⋅

211

Figure 4.3: The five vertices in U connected to the lower vertex with label “211” in V .

Both the cones and the funnels provide a partition of the set of edges. It is instantly clear that each funnel is finite, and so the funnels are not as important as the cones. The structure of a cone is more interesting than that of a funnel, as the following proposition shows. Proposition 4.36. Let p be any prime. Accordingly, in the graph 𝒢 , the cone ℰp is peaking at the upper vertex (p, 1), and for each natural number n, it contains an edge e = ((p, 1), (q, 0)) labeled νp (q − 1) ≥ n. In particular, ℰ contains infinitely many edges. Proof. There is nothing to prove—this is just a translation of Lemma 4.3 into the language of the mastergraph 𝒢 . The labels of the sloping edges in Figures 4.2 and 4.3 are all “1” with the exception of 2–5 for which it is “2”. All applications of prime graphs which follow in the course of this text are subgraphs of this mastergraph.

̃× and the prime mastergraph 4.3.2 The structure of ℤ Since for any periodic locally compact abelian group A we have a canonical surjective ̃ × → SAut(A) we need explicit information on the primary structure—or morphism ζ : ℤ ̃×. p-Sylow structure—of ℤ We recall that ℰ is the set of all edges of the mastergraph 𝒢 = (U, V, ℰ , λ). We start the indexing by attaching to each edge e = ((p, 1), (q, 0)) ∈ ℰ a profinite group 𝕊e being, ̃ up to a natural isomorphism, a subgroup of ℤ.

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| 117

Definition 4.37. For each edge e ∈ ℰ from p to q we set ℤ2 ⊕ ℤ(2), { { { 𝕊e = {ℤp , { { νp (q−1) ), {ℤ(p

if p = q = 2,

(4.12)

if 2 < p = q, if p < q.

× ̃× = ∏ We noted in (4.11) that ℤ p∈π ℤp and in Lemmas 4.13 and 4.16 a procyclic p-group occurs precisely as a subgroup of some 𝕊e for an edge e with upper vertex p. ̃ × is represented by the cone ℰ peaking in p, i. e., Therefore the p-Sylow subgroup of ℤ

̃ × )p ≅ (∏ ℤ× ) = ∏ 𝕊e , (ℤ q

(4.13)

p 󳨃→ ℰp : π → 𝒞

(4.14)

q∈π

p

e∈ℰp

is a bijection from the set of primes to the set 𝒞 of cones such that 𝒞 = ⋃p∈π ℰp in the mastergraph. Taking these matters and Proposition 4.4 into account, we can summarize as follows. ̃ × of units of the universal procyclic compactification ℤ ̃ Theorem 4.38. (i) The group ℤ of the ring of integers ℤ is the product ̃ × ≅ ∏ 𝕊e ℤ

(4.15)

e∈ℰ

extended over the set ℰ of all edges of the mastergraph, where 𝕊e is the profinite group given in (4.12). (ii) Its p-Sylow subgroup is the subproduct extended over the cone peaking in p, i. e., ℤ ⊕ ℤ(2) ⊕ ∏q>2 ℤ(2ν2 (q−1) ), ̃ × ) = ∏ 𝕊e ≅ { 2 (ℤ p ℤp ⊕ ∏q>p ℤ(pνp (q−1) ), e∈ℰp

if p = 2, otherwise.

(4.16)

(iii) For each p ∈ π fixed, ̃ × ) ≅ ℤp ⊕ Tp , (ℤ p

̃×) where Tp = tor(ℤ p

(4.17)

and where Tp contains a ℤp -submodule algebraically isomorphic to ℤ(ℕ) p whose clon sure is isomorphic to ∏n∈ℕ ℤ(p ). ̃ For each prime p, define Let T = tor(ℤ). ℤℙp = ∏ ℤ(pn ), n∈ℕ

ℤℙ = ∏ ℤℙp = ( p∈π

∏

(p,n)∈π×ℕ

ℤ(pn )).

(4.18)

118 | 4 Scalar automorphisms and the mastergraph ̃ ̃ (ℕ) . Corollary 4.39. (i) ℤℙ contains a dense copy of the torsion-free ℤ-module M := ℤ × ̃ contains a copy of M. (ii) The closure T of the torsion subgroup of ℤ Proof. (i) The compact abelian group ℤℙp contains a dense copy of ℤ(ℕ) (see Thep

which orem 4.38 (iii)). Hence ℤℙ = ∏p∈π ℤℙp contains a dense copy of ∏p∈π ℤ(ℕ) p ̃ (ℕ) ≅ ∏ ℤ and this copy is still dense in ℤℙ. contains a copy of ℤ p∈π p (ii) Theorem 4.38 (iii) implies that for each prime, Tp contains a copy of ℤℙp . Hence T contains a copy of ℤℙ.

4.3.3 The Sylow decomposition of ℤ(n)× indexed by 𝒢 We record n = ∏p|n pνp (n) (finite product: almost all νp (n) ≠ 0 only if p|n) and accordingly ℤ(n) = ∏p|n ℤ(pνp (n) ). Hence ℤ(n)× = ∏p|n ℤ(pνp (n) )× , and it suffices to recall the case that n = pm . This we assume for the remainder of this section, and we fix a prime p. Here we have ℤ(pm ) = ℤp /pm ⋅ℤp . Let μ: ℤp → ℤp denote the scalar endomorphism given by μ(x) = pm x. Then μ

0 → ℤp 󳨀󳨀󳨀󳨀→ ℤp → ℤ(pm ) → 0 is exact and μ induces a quotient morphism μ× : ℤ×p → ℤ(pm )× . We recall that the morphism ℤp → ℤp /pℤp ≅ GF(p) maps Cp of Lemmas 4.13 and 4.16 faithfully because pm ℤp ⊆ pℤp unless p = 2 and m ≤ 2. In the latter situation pm = 2 or = 4, in which case we have ℤ(2)× = {1}, respectively, ℤ(4)× = {±1}. If p > 2, then we know that exp: (pℤp , +) → (1 + pℤp , ×) is an isomorphism, whence by applying μ exp: ( Since

pℤp pm ℤp

pℤp

pm ℤp

, +) → (μ(1 + pℤp ), ×) is an isomorphism.

≅ ℤ(pm−1 ) in view of Lemma 4.13 we have ℤ(pm ) ≅ ℤ(pm−1 ) ⊕ ℤ(p − 1). ×

Analogously, for p = 2 and m > 2, from Lemma 4.16 we obtain ℤ(2m ) ≅ ℤ(2m−2 ) ⊕ ℤ(2). ×

Summarizing, we have the following lemma.

(4.19)

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| 119

Lemma 4.40. The group of units of ℤ(pm ) is {0}, { { { { { {ℤ(2), × ℤ(pm ) ≅ { {ℤ(2m−2 ) ⊕ ℤ(2), { { { { m−1 {ℤ(p ) ⊕ ℤ(p − 1),

if pm = 2,

if pm = 4, if p = 2, m > 2,

(4.20)

if p > 2.

We may use 𝒢 as index set for describing the p-Sylow decomposition of A = ℤ(pm )× as follows. We index subgroups 𝕊e ≤ A by attaching again to each edge e = ((p, 1), (q, 0)) ∈ ℰ ̃×. a profinite group 𝕊e being, up to a natural isomorphism, a subgroup of ℤ Definition 4.41. For each edge e ∈ ℰ from p to q we set {0}, if pm = 2 or q > pm , { { { { { { ℤ(2), if pm = 4 and p = q = 2, { { { 𝕊e = {ℤ2 ⊕ ℤ(2), if p = q = p = 2, { { { { ℤ(pm−2 ), if 2 < p and q ≤ p, { { { { νp (q−1) ), if p < q ≤ p. {ℤ(p

(4.21)

With this indexing we can formulate the following theorem. Theorem 4.42. For a fixed prime p and a fixed natural number m, we have the following. (i) The group ℤ(pm )× of units of the universal cyclic group ℤ(pm ) is ℤ(pm ) = ∏ 𝕊e ×

e∈ℰ

(4.22)

extended over the set ℰ of all edges of the mastergraph, where 𝕊e is the profinite group given in (4.21) above. (ii) Its p-Sylow subgroup is the subproduct extended over the cone peaking in p, i. e., ℤ(4) ⊕ ℤ(2) ⊕ ⨁p≥q>2 ℤ(2ν2 (q−1) ), if p = 2, × (ℤ(pm ) )p = ∏ 𝕊e ≅ { otherwise. ℤ(pm−2 ) ⊕ ⨁p≥q>p ℤ(pνp (q−1) ), e∈ℰp

(4.23)

4.3.4 The structure of SAut(A) and its prime graph 𝒢(A) Now let A be a periodic locally compact abelian group; the Sylow structure of SAut(A) ̃ × → SAut(A) of Proposition 4.23, is now easily discussed: The quotient morphism ζ : ℤ preserving the Sylow structures, and the structure of SAut(A) described so far in Theorem 4.25 allow a precise description of the Sylow structure of SAut(A).

120 | 4 Scalar automorphisms and the mastergraph We associate with A the bipartite graph 𝒢 (A) = (U, V, ℰ (A), λ) with U and V as in the mastergraph and with ℰ (A) = {e ∈ E : e = ((p, 1), (q, 0)) such that SAut(Aq )p ≠ {idA }},

and for fixed p we define ℰp (A) to be the set of all edges in 𝒢 (A) from p to q. Define 𝕊e (A) := SAut(Aq )p and let ℱq (A) be the set of all edges in 𝒢 (A) from p to q (i. e., terminating in q). Finally, for e ∈ ℰ (A) from p to q the label is 0, if p = q, λ(e) = { νp (q − 1), if p|(q − 1).

(4.24)

Now let A be a periodic locally compact abelian group; the Sylow structure of ̃ × → SAut(A) of ProposiSAut(A) is easily discussed: The quotient morphism ζ : ℤ tion 4.23, preserving the Sylow structures, and the structure of SAut(A) described so far in Theorem 4.25 allow a precise description of the Sylow structure of SAut(A). Theorem 4.43 (The Sylow Structure of SAut(A)). Let A be a periodic locally compact abelian group and SAut(A) = ∏p∈π SAut(A)p the p-primary decomposition of the profinite group SAut(A) = ∏e∈ℰ(A) 𝕊e . Then (i) The p-primary decomposition of SAut(Aq ) is (additive notation assumed) ∏ SAut(Aq )pe = ∏ 𝕊e (A),

e∈ℱq

e∈ℱq

and this group is isomorphic, in the case p = 2, to {0}, if A2 has exponent ≤ 2, { { { r−2 ℤ(2 ) ⊕ ℤ(2), if A2 has finite exponent 2r > 2, { { { if A2 has infinite exponent, {ℤ2 ⊕ ℤ(2), and, in the case p > 2, to {ℤ(qr−1 ) ⊕ ⨁e∈ℱq ℤ(peλ(e) ), if Aq has finite exponent qr , { ℤ ⊕ ⨁e∈ℱq ℤ(pλ(e) if Aq has infinite exponent. e ), { q

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| 121

(ii) The structure of the p-primary component SAut(A)p of SAut(A) (in additive notation) is ∏ SAut(Aqe )p

e∈ℰp

= ∏ 𝕊e (A) e∈ℰp

if p = 2 and A2 has exponent ≤ 2, ∏e∈ℰp ℤ(2λ(e) ), { { { { r−2 λ(e) r { { {ℤ(2 ) ⊕ ℤ(2) ⊕ ∏e∈ℰp ℤ(2 ), if p = 2 and A2 has fin. exp. 2 >2, { { ≅ {ℤ2 ⊕ ℤ(2) ⊕ ∏e∈ℰp ℤ(2λ(e) ), if p = 2 and A2 has inf. exponent, { { { r−1 λ(e) { ℤ(p ) ⊕ ∏e∈ℰp ℤ(pe ), if 2 < p and Ap has exponent pr , { { { { λ(e) if 2 < p and Ap has infinite index. {ℤp ⊕ ∏e∈ℰp ℤ(pe ), This theorem illustrates the usefulness of the prime graph 𝒢 (A) which elucidates the fine structure of SAut(A). The edges of 𝒢 (a) serve as the index set for a product decomposition SAut(A) = ∏e∈ℰ(A) 𝕊e which refines the product decomposition SAut(A) = ∏p∈π SAut(Ap ) which is induced by the p-primary (or Sylow) decomposition A = ∏loc p∈π (Ap , Cp ). Theorem 4.43 tells us that in the product decomposition SAut(A) = ∏e∈ℰ(A) 𝕊e each partial product over a cone in 𝒢 (A) yields a Sylow subgroup.

4.3.5 The structure of a scalar action and its prime graph Assume now that Γ is a multiplicatively written periodic locally compact abelian group acting by scalar morphisms on an additively written periodic locally compact abelian group A, that is, we assume a morphism α: Γ → SAut(A). In the main body of this text, Γ will be rather special. With these data we now associate a subgraph 𝒢α = (U, V, ℰα ) ⊆ 𝒢

of the mastergraph as follows: ℰα = {e = ((p, 1), (q, 0)) : α(Γp )|Aq ≠ idAq }.

Definition 4.44. The bipartite subgraph 𝒢α of the mastergraph 𝒢 is called the bipartite prime graph associated with α.

122 | 4 Scalar automorphisms and the mastergraph We shall briefly call 𝒢α the prime graph of α. We define the cones ℰp and funnels ℱq in analogy to what we defined before. For each edge e = ((p, 1), (q, 0)) ∈ ℰα we have a morphism αe : Γp → SAut(Aq )p ≅ 𝕊e (A) and for each prime p we have a morphism αp : Γp → SAut(A)p = ∏ 𝕊e (A) e∈ℰp

between the p-primary components of Γ and SAut(A) according to Theorem 4.43 (ii). Therefore, the family (αe )e∈ℰ(α) completely determines α uniquely since any morphism between periodic abelian groups is determined by its restriction and corestriction to their primary components. As SAut(Aq ) is isomorphic to the group of units ℛ(Aq )× according to Theorem 4.25 we obtain an invertible element for each edge e ∈ ℰα a scalar-valued morphism χe : Γpe → ℛ(Aqe )× such that αe (g)(a) = χe (g)⋅a for all g ∈ Γpe and each a ∈ Aqe . In this sense the prime graph 𝒢α is a weighted bipartite graph in which to every edge from e = ((p, 1), (q, 0)) ∈ ℰα an ℛ(Aqe )× -scalar-valued character χe is attached. If, by way of example, Γ is monothetic, then its action is completely determined by the action of a generator, and that means by an element ρ ∈ SAut(A). Such a ρ is ̃ × = ∏ ℤ× acting componentwise by multiplication by a suitable scalar r = (rp ) ∈ ℤ p p

scalar multiplication on A = ∏loc p (Ap , Cp ). (Cf. Theorem 3.3.) Therefore the bipartite graph attached to this situation is 𝒢r with Rr containing all edges e = ((1, p), (0, q)) ̃ × and a ∈ Aq . The details are explained in the such that rp ⋅a ≠ a for some rp ∈ ℤ p following example. Example 4.45. Recall that an edge e in the cone ℰp with vertex p = pe in 𝒢r runs down from p to qe so that ℤ×p = ∏ (ℤ×qe )p . e∈ℰp

Therefore, rp = (re )e∈ℰp ∈ ∏ (ℤ×qe )p e∈ℰp

for being a p-element, while r = (re )e∈ℰα ∈ ∏ (ℤqe × )p . e∈ℰα

e

Then the scalar r acts on elements a = (aq )q ∈ A = ∏loc q (Aq , Cq ) (according to Theorem 3.3) as a 󳨃→ r⋅a, defined by r⋅a =

∏

q∈π(A), e∈ℱq

(re ⋅aq )q .

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| 123

This is well defined due to the fact that each funnel ℱq is finite. In particular, since each (ℤ×qe )pe is a procyclic pe -group for all e ∈ R we can take for re a generator of this group. ̃ and let Γ = ⟨ρ⟩ be the universal procyclic group acting on If we finally let A be ℤ A via α: Γ → SAut(A), α(ρ)(a) = r⋅a, then the prime graph 𝒢r of this action is the full mastergraph 𝒢 . For our later discourse, the most frequent case of a scalar action of a locally compact abelian group Γ on a periodic locally compact abelian group A occurs if A is a closed normal subgroup of a topological group G in which case Γ := G/A acts in the obvious fashion on A. This leads us to the following definition. Definition 4.46. If (G, A) is a pair consisting of a topological group G and a closed normal subgroup A, then we call it a special extension of A if G is a locally compact group and the equivalent conditions of Proposition 4.9 are satisfied. In this case we shall denote the bipartite prime graph 𝒢α associated with α : G/A → SAut(A) by 𝒢 (G, A) and call it the prime graph of the special extension (G, A). From Proposition 4.9 we recall that we have a morphism ρ: G → ℛ(A)× , which provides for each g ∈ G a scalar r = ρ(g) such that for all a in A with have gag −1 = r⋅a. In particular, for every edge e ∈ ℰ (G, A) in the prime graph of the special extension (G, A) from some p to some q, by the very existence of e there is a p-element g ∈ Gp and a scalar 1 ≠ r := ρ(g) ∈ (ℛ(Aq )× )p . Set CA (g) = {a ∈ A : [g, a] = 1}. With this notation we now have the following result. Theorem 4.47. Let (G, A) be a special extension of a periodic locally compact abelian group and e ∈ ℰ (G, A) an edge in its prime graph from p to q. Accordingly, the following conclusions hold: (i) If p ≠ q, then all of Aq consists of commutators. In particular, Aq ⊆ G󸀠 . (ii) If p = q, then Ap ∩ CA (g) is a torsion group with an exponent dividing qm for some m ∈ ℕ. Proof. According to Definition 4.44 the presence of the edge e ∈ ℰ (G, A) means that there is a p-element g ∈ Gp such that 1 ≠ r = ρ(g). Case (i). Since e is sloping, p < q. By Theorem 4.25 we know that (ℛ(Aq )× )p is a cyclic group of order pλ(e) = pνp (q−1) . We claim that 1 − r is a unit in the ring ℛ(Aq ) of scalars which is isomorphic to ℤq or a quotient ring thereof depending as Aq has infinite or finite exponent. By way of contradiction suppose that r − 1 is not a unit. Since ℤ×q = ℤq \ qℤq , there is an element

124 | 4 Scalar automorphisms and the mastergraph u ∈ ℛ(Aq ) such that 1 − r = qu. Then r = 1 − qu ∈ 1 + qℛ(Aq ) which, according to the structure of ℤ×q in (4.8), respectively, of ℤ(qm )× in (4.20), is the q-Sylow subgroup of ℛ(Aq )× . But r is a p-element with p < q and this is a contradiction. Now let a ∈ Aq . For the purpose of this proof we write G additively. Then the commutator of g and a is [g, a] = ρ(g)(a) − a = r⋅a − a = (r − 1)⋅a. Since 1 − r is invertible, we set b = (r − 1)−1 ⋅a ∈ Aq and obtain a = ρ(g)(b) − b = g + b − g − b = [g, b]. This shows that every element of Aq is a commutator and thus proves the assertion of case (i). Case (ii). The scalar r now is a p-element in the p-Sylow subgroup of ℛ(A)× and thus is of the form r = exp pv for some by Theorem 4.25 (c). Consequently, there is a unique unit s ∈ ℤ×p and a unique natural number m such that r − 1 = pm s. Now let z ∈ Ap ∩ CA (g). Then r⋅z = gzg −1 = z, i. e., (r − 1)⋅z = 0 and so pm ⋅z = s−1 (r − 1)⋅z = 0. Thus Ap ∩ CA (g) is a torsion group with an exponent e such that e|pm . In the preceding proof, we considered the action of a p-element scalar r on an abelian locally compact q-group A. For recalling easily the structure of the group of units of ℤq for a prime q, we remember, e. g., the notation of Proposition 4.17, i. e., (1 + 4ℤ2 , ×), ℙq := { 1 + qℤq , ×),

if q = 2, otherwise,

({1, −1}, ×), if q = 2, Rq := { Cq−1 , otherwise,

(4.25)

where Cq−1 denotes the cyclic group of the (q − 1)th roots of unity contained in ℤq . The exponential function exp: (qℤ, +) → ℙq (respectively, exp: (4ℤ, +) → ℙq if q = 2) yields an isomorphism of groups so that ℙq is a q-group. We then recall that ℤ×q = ℙq Rq ≅ ℙq × Rq . (These things were reviewed in [57], Section 2.1, notably in Lemmas 2.1 and 2.2.) So ℙq is the q-primary component of ℤ×q ; more specifically, the primary decomposition of ℤ×q is ℤ×q = ℙq × ∏ (Rq )r , r|(q−1)

where we recall Rq from equation (4.25). Definition 4.48. A vertex in a prime graph 𝒢α is called isolated if it is not the endpoint of any edge. Remark 4.49. In the circumstances of Theorem 4.47 we have ℙq , if e is vertical, 𝕊e (Aq ) = { (Rq )p , otherwise.

4.3 A bipartite graph for scalar action on a periodic locally compact abelian group

| 125

In either case, the q-vertex of the prime graph 𝒢 (G, A) of the special extension (G, A) graph is not isolated. We conclude the section with a simple observation on the units in ℤp . Since ℤp is a local ring extension of ℤ with the unique maximal ideal m = pℤp with quotient field GF(p) it follows that every element u ∈ ℤp can be uniquely presented in the form u = α + pv with 0 ≤ α ≤ p − 1 and v ∈ ℤp . (See, e. g., [67].) Lemma 4.50. Let u ∈ ℤ×p . Then there is an s ∈ ℕ with 1 ≤ s ≤ p − 1 and a v ∈ ℤ×p with u = (1 + ps )v. Proof. As has been said, since u is a unit there is a natural number 1 ≤ α ≤ p − 1 such that u ∈ α + pℤp . Then, as ℤp is a local ring and all its proper ideals are of the form ps ℤp , there is a maximal s ∈ ℕ with u − α ∈ ps ℤp which implies that u − α = ps v for some element v ∈ ℤp . Then v cannot be contained in pℤp , else u − α ∈ ps+1 ℤp , contradicting the maximality of s. Note that of course s ≥ 1. Hence u = α + ps w for some unit w ∈ ℤ×p . Noting that (1 + ps ) ∈ ℤ×p shows that we can solve the equation u = (1 + ps )v in ℤ×p , as claimed.

5 Inductively monothetic groups Under this name we re-introduce topological rank-1 groups originally proposed by V. S. Čarin in [15] and also studied by M. S. Khan in [65] (who termed them “t-cyclic)”— a generalization of quasicyclic groups. If every finitely generated subgroup of a group G satisfies a property 𝒫 , then G is said to be inductively 𝒫 . Notation 5.1. When G is any topological group and 𝒫 is the property of being monothetic we have now defined what it means that G is an inductively monothetic group, namely, G is an inductively monothetic group if and only if for every finite subset F of G there is an element g ∈ G such that ⟨F⟩ = ⟨g⟩. Obviously, a closed subgroup of an inductively monothetic group is an inductively monothetic group. The relation between the properties of being “monothetic” and “inductively monothetic” is not as obvious as one might surmise without looking very closely: The torus G = 𝕋2 , 𝕋 = ℝ/ℤ, is monothetic (see, e. g., [54], Theorem 12.22.) but the finitely generated subgroup ( 21 ⋅ℤ/ℤ)2 ≅ ℤ(2)2 is not monothetic, so G is not inductively monothetic. In the discrete group G = ℤ(2∞ ) = 21∞ ⋅ℤ/ℤ every finite subset is contained in a cyclic subgroup 21n ⋅ℤ/ℤ ≅ ℤ(2n ); thus G is an inductively monothetic group, but is far from being monothetic. After our classification Theorem 5.15 we shall know that every totally disconnected monothetic group, equivalently, every procyclic group, is inductively monothetic, while inductively monothetic groups containing nondegenerate connected subsets are necessarily topologically one-dimensional compact connected monothetic groups. Proposition 5.2. A locally compact abelian p-group G is inductively monothetic if and only if it has p-rank 1. Proof. This is an immediate consequence of Lemma 3.91, the definition of p-rank (see Definition 3.6 and Theorem 3.97). It is a well-known fact that any finitely generated discrete abelian group in which any two subgroups intersect nontrivially must be cyclic. This is an immediate consequence of the structure theorem of finitely generated abelian groups. Lemma 5.3. Let G be a locally compact abelian compactly ruled group. If any nontrivial closed subgroups intersect nontrivially, then G is an inductively monothetic compact p-group for some prime p. Proof. By Braconnier’s Theorem 3.3 G is a local product loc

G = ∏ (Gp : Cp ), p∈π(G)

https://doi.org/10.1515/9783110599190-005

128 | 5 Inductively monothetic groups for some compact C = ∏p∈π(G) Cp , and since for p ≠ q the corresponding primary groups intersect trivially it follows that G is a p-group. Let C be an arbitrary compact subgroup of G. Then also any nontrivial closed subgroups of C intersect nontrivially. Pick a ∈ C \ pC; then ⟨a⟩ ∩ pC = p⟨a⟩, so that as a consequence of Corollary 3.39, ⟨a⟩ must be a direct summand of G. Since by assumption C cannot split nontrivially deduce that C = ⟨a⟩ is procyclic. In particular, whenever F is a finite subset of G, our consideration applies to C := ⟨F⟩ and shows that the latter subgroup is a monothetic p-group and therefore G is an inductively monothetic p-group. Corollary 5.4. Let A be a locally compact abelian p-group containing properly an open compact subgroup U of exponent p. If A is not inductively monothetic, then there are elements a ∈ A \ U and 0 ≠ b ∈ U with ⟨a⟩ ∩ ⟨b⟩ = {0}. Proof. If U is cyclic, then any two nontrivial subgroups of A have intersection U. Then, by Lemma 5.3, G is inductively monothetic, contrary to our assumptions. Therefore U is not cyclic and, whenever a ∈ A\U, there must exist b ∈ U \A. Clearly ⟨a⟩ ∩ ⟨b⟩ = {0}. In agreement with all this we say that a discrete torsion p-group has p-rank 1 if its p-primary component is isomorphic to ℤ(pn ) for some n ∈ {1, 2, . . . , ∞}. (Cf. [54], Definition A1.20 and subsequent remarks.)

5.1 Classifying inductively monothetic subgroups Theorem 5.15 provides a complete description of inductively monothetic groups. An important consequence is the gamma Theorem 5.26, dealing with inductively monothetic subgroups of SAut(G). For purely technical reasons we say that a topological group is of class (I) of class (II) of class (III)

if it is discrete, if it is infinite compact, and if it is locally compact but is neither of class (I) nor of class (II).

Lemma 5.5. If G is an inductively monothetic group of class (I), then it is isomorphic to a subgroup of ℚ or else of ℚ/ℤ, that is, if and only if it is either torsion-free of rank 1 or a torsion group of p-rank 1 for all primes p. Proof. This is an exercise in view of the fact that a finitely generated abelian group is a direct sum of cyclic ones which then must be cyclic. If G is class (I) and is the directed union of subgroups which are inductively monothetic groups, then G is an inductively monothetic group.

5.1 Classifying inductively monothetic subgroups | 129

Lemma 5.6. If G is an inductively monothetic group of class (II), then it is monothetic and it is either connected of dimension 1, or it is infinite profinite. Proof. By Proposition 2.42 of [54], a compact abelian group G is a Lie group iff it is of the form 𝕋n × F with a finite group F. For n ≥ 1, the monothetic torus 𝕋n contains the subgroup ℤ(2)n which is an inductively monothetic group only if n = 1. If C ⊆ F is a nondegenerate cyclic subgroup of F and n = 1, then 𝕋 contains an isomorphic copy C∗ of C and so G contains C∗ × C which fails to be an inductively monothetic group. Thus G is a class (II) group and a Lie group iff it is isomorphic to 𝕋 or is a finite class (I) group, in which case its p-rank is 1 for all primes p dividing the order of F. Thus G is a ̂ is a finitely generated Lie group of class (II) and an inductively monothetic group iff G discrete inductively monothetic group. Now let G be an inductively monothetic group of class (II). Then it is a strict projective limit of its Lie group quotients which are inductively monothetic groups of class ̂ is the directed union of its finitely generated subgroups which are induc(II). Hence G ̂ is an inductively monothetic group. tively monothetic groups of class (I), and thus G ̂ By Lemma 5.5, G is a subgroup of ℚ or of ℚ/ℤ. Now, by duality, G is of the kind asserted (cf. [54], 822.ff.). The proof has shown that an infinite compact abelian group G is an inductively ̂ is an infinite inductively monothetic group of monothetic group of class (II) iff G class (I). Lemma 5.7. An inductively monothetic group of class (III) is periodic. Proof. Suppose ℝ were an inductively monothetic group. But ℝ is generated by 1 and √2; thus ℝ would have to be monothetic which it is not. Thus an inductively monothetic group of class (III) does not contain any vector subgroup and so the subgroup E in Theorem 7.57 of [54] is trivial and G has a compact open subgroup. Since G is not discrete, there is an open infinite compact subgroup U which is an inductively monothetic group. Suppose it were not zero-dimensional. Then by Lemma 5.6 it would be isomorphic to a one-dimensional compact connected group. Being divisible and open, it would be an algebraic and topological direct summand; G being an inductively monothetic group would imply G = U, which is impossible since G is of class (III). Thus U and hence G is zero-dimensional. Suppose G is a class (III) inductively monothetic group for which G ≠ comp(G). Then by [54], Corollary 4.5, there would be an infinite cyclic discrete group D ≅ ℤ and for every compact open subgroup U the open sum U + D is direct. Now G being an inductively monothetic group implies that U has to be trivial. Then G would be discrete, but it is of class (III). Therefore G = comp(G) and thus G is periodic. Now Theorem 3.3 reduces the classification of class (III) inductively monothetic groups to the case that G is a p-group.

130 | 5 Inductively monothetic groups Lemma 5.8. Let G be a p-group which is a class (III) inductively monothetic group. Then G ≅ ℚp . Proof. Proposition 5.2 implies that such G is a torsion-free p-rank 1 group. Since G is neither compact nor discrete it follows from Theorem 3.97 that G ≅ ℚp topologically and algebraically. Definition 5.9. (i) A topological group G is called Π-procyclic if: (a) it is locally compact abelian periodic and (b) each p-primary component is a procyclic p-group. (ii) We shall call G a standard inductively monothetic group iff it is isomorphic to a local product loc

∏(Mp , Cp ), p

where Mp is either ℚp , ℤ(p∞ ) (discrete), or a procyclic p-group and where Cp is a compact open subgroup of Mp . Remark 5.10. (i) A compact group is Π-procyclic iff it is procyclic. (ii) A periodic inductively monothetic group G is Π-procyclic iff it is reduced, that is, G does not contain any nontrivial divisible subgroups. (iii) A discrete group is Π-procyclic iff it is a reduced subgroup of the discrete group ℚ/ℤ (that is, each of its p-primary components is finite).

5.1.1 The build-up of Π-procyclic groups from open procyclic subgroups In the following brief discourse we show that a Π-procyclic group can, in a systematic way, be represented as an ascending union of compact open procyclic subgroups, and we exhibit the respective generators in a systematic way. Remark 5.11. The p-primary components of a Π-procyclic group are of the form (M, C) with M ≅ ℤp or M ≅ ℤ(pn ) = ℤ/pn ℤ for some n and C = ν⋅M for some nonnegative integer ν (with ν|pn in the second case). Thus if g is a generator of M (such as g = 1 ∈ ℤp ) then g ν is a generator of C. In the following we let p1 = 2, p2 = 3, . . . be the sequence of primes in increasing order. Let P = ∏loc n∈ℕ (Ppn , Cpn ) be a Π-procyclic group allowing for the possibility that Cpn = Ppn . Then C = ∏n∈ℕ Cpn is a compact open subgroup of P. Definition 5.12. Set ν(n) = |Ppn /Cpn | and let gn be a generator of Ppn . We define P[m] = {(xpk )k∈ℕ ∈ P : (∀n ≥ m) xpn ∈ Cpn }.

(1)

5.1 Classifying inductively monothetic subgroups | 131

Equivalently, C, if m = 1, P[m] = { Pp1 × ⋅ ⋅ ⋅ × Ppm−1 × ∏n≥m Cpn , if m > 1.

(2)

The following information on the subgroup structure of a Π-procyclic group is now rather immediate. A point of notation: For a subgroup X of a multiplicatively written abelian group and a natural number s we write X [s] := {xs : x ∈ X}. Proposition 5.13. In these circumstances, the following statements hold:

(i) Cpn = ⟨gnν(n) ⟩, (ii) C = ⟨c⟩ for c := (gnν(n) )n∈ℕ , (iii) P[1] = C and for all m > 1 the subgroup P[m] is procyclic with generator ν(m) ν(m+1) , gm+1 , . . . ); g[m] := (g1 , . . . , gm−1 , gm

in particular, g[m]ν(1)⋅⋅⋅ν(m−1) = c, (iv) if m < n, then g[m] = g[n]ν(m)⋅⋅⋅ν(n−1) ,

(†)

(v) each P[m] is compact open in P and P is the ascending union of the P[m], (vi) for all m we have P[m + 1][ν(m)] = P[m]; notably, if m < n, then P[n][ν(m)⋅⋅⋅ν(n−1)] = P[m].

(‡)

Proof. (i) This is a consequence of the known structure of ℤpn and its homomorphic images. (ii) This follows from (i) and C = ∏n∈ℕ Cpn . Statements (iii), (iv), and (v) are immediate from (2). (vi) Since multiplying an additively written p-group by q is an automorphism for different primes p and q we know that x → xν(m) induces an automorphism on ∏loc n∈ℕ,n=m ̸ (Ppn , Cpn ) while it maps the factor Ppm onto its open subgroup Cpm . In view of (2) this proves the claim. 5.1.2 Inductively monothetic groups and divisibility From Lemma 5.5 we know that an inductively monothetic group D of class (I) is divisible if it is either isomorphic to ℚ or to a direct sum ⨁p∈ℙ󸀠 ℤ(p∞ ) for a set ℙ󸀠 of prime numbers. Lemma 5.6 says that an inductively monothetic group of class (II) is compact connected monothetic since a compact group is divisible iff it is connected (see [54], Theorem 9.35, p. 479). Lemmas 5.7 and 5.8 imply that the p-primary component of a divisible inductively monothetic group of class (III) is isomorphic to ℚp or

132 | 5 Inductively monothetic groups to ℤ(p∞ ). Every abelian group G has a unique largest divisible subgroup D which is a direct summand (see, e. g., [54] A1.31, p. 686 and A1.36, p. 691). If a periodic locally compact abelian group G has the property that each p-primary component Gp has p-rank ≤ 1 as is the case of a torsion inductively monothetic group of class (I) or an inductively monothetic group of class (III), then its maximal divisible subgroup D is a direct factor of G and is isomorphic to ∏p∈ℙ󸀠 (Mp , Cp ) with Mp = ℚp and Cp = ℤp or Mp ≅ ℤ(p∞ ); the complementary factor ∏loc p∉ℙ󸀠 (Mp , Cp ) is Π-procyclic. So we have the following.

Proposition 5.14. A periodic inductively monothetic locally compact group G is a direct product D×P, where D is isomorphic to a local product ∏loc p∈δ (Mp , Cp ) for a set δ of primes ∞ p with (Mp , Cp ) = (ℚp , ℤp ) or (Mp , Cp ) with Mp = ℤ(p ), and where P is a Π-procyclic group whose p-primary components are zero for p ∈ δ. So, the summarizing of Lemmas 5.5, 5.6, 5.7, and 5.8 and Proposition 5.14 yields the following. Theorem 5.15 (Classification theorem of inductively monothetic groups). Let G be an inductively monothetic locally compact group. Then G is either – a one-dimensional compact connected abelian group, or – totally disconnected and isomorphic to exactly one of the following groups: (1) a subgroup of the discrete group ℚ, (2) a standard inductively monothetic locally compact group loc

∏(Mp , Cp ). p

All inductively monothetic groups are sigma-compact. The groups of connected type in Theorem 5.15 are monothetic; other types may or may not be monothetic. In view of Corollary 1.6, an inspection of each of the three conditions in Theorem 5.15 yields at once the following. Corollary 5.16. For a locally compact inductively monothetic group G the following conditions are equivalent: (1) G = comp(G), (2) G is compactly ruled, (3) G is not isomorphic to a subgroup of the discrete group ℚ. Also, the following two conditions are equivalent: (4) G is periodic, (5) G is not connected and not isomorphic to a subgroup of the discrete group ℚ.

5.1 Classifying inductively monothetic subgroups | 133

As a consequence of these remarks and of Weil’s lemma [54, Proposition 7.43] we record the following. Corollary 5.17. Let the inductively monothetic group G contain a discrete subgroup D ≅ ℤ. Then G is discrete and torsion-free of rank one. The next fact has already been observed by M. S. Khan in [65]. Corollary 5.18. The class ℐℳ is closed under the passage from a locally compact abelian group to its character group. Proof. Let G be in ℐℳ. Assume, firstly, that G is compact connected one-dimensional. A compact abelian group is connected and one-dimensional if and only if its character group is torsion-free and is of rank one, which is equivalent to saying that it is isomor̂ is isomorphic to a phic to a subgroup of ℚ (see, e. g., [54, Theorem 8.22 ff.]). Thus G subgroup of ℚ and hence belongs to ℐℳ by Theorem 5.15. ̂ Assume, secondly, that G is discrete and is isomorphic to a subgroup of ℚ; then G by the same reasoning is compact connected and one-dimensional. Now assume, thirdly, that loc

G ≅ ∏(Mp , Cp ), p

(1)

̂p is isomorwhere Mp is isomorphic to (a) ℤ(pmp ), (b) ℚp , (c) ℤ(p∞ ), or (d) ℤp . Then M ∞ phic to Mp in cases (a) and (b), to ℤp in case (c) and to ℤ(p ) in case (d). Further, the subgroup Cp ⊆ Mp is compact and open in Mp , and it is open iff Mp /Cp is discrete. Then ̂p is isomorphic to the character group of Mp /Cp and is therefore its annihilator Cp⊥ in M ⊥ ̂ compact and Mp /Cp is isomorphic to the character group of Cp and is therefore diŝp . So the local crete. (See, e. g., [54, Theorem 7.64.].) Thus C ⊥ is compact and open in M p

product

̂p , C ⊥ ) L = ∏(M p p

(2)

is well defined and is inductively monothetic by Theorem 5.15. However, by Braconnier’s duality Lemma 3.82, the local products in (1) and (2) are character groups of each ̂ This completes the proof of the corollary. other. Hence L ≅ G. This corollary expresses the fact that the class ℐℳ of inductively monothetic locally compact abelian groups is closed under duality. This is not at all obvious from its original Definition 5.1. We remark that the class of monothetic locally compact groups is far from fitting into such a pattern: Recall that a locally compact abelian group G is ̂ is discrete and isomorphic to a subgroup of ℝ/ℤ monothetic if either it is ≅ ℤ or else G (see, e. g., [54, Proposition 7.43 and Example 8.75]).

134 | 5 Inductively monothetic groups Lemma 5.19. Let ϕ be a continuous morphism from an inductively monothetic p-group G onto a nonsingleton compact group Q. Then G must be compact and monothetic p-group. Proof. By the classification Theorem 5.15 the p-group G can only be isomorphic to a group Mp where according to Definition 5.9 (ii) Mp is (a) ℚp , (b) ℤ(p∞ ), or (c) compact procyclic. In cases (a) and (b), G is divisible and so is any quotient of G. A compact divisible group is connected; but Q as a quotient of a p-group is a p-group and therefore is totally disconnected. A compact group that is simultaneously connected and totally disconnected is singleton. Hence cases (a) and (b) do not apply, and so G is compact procyclic. Again inspect the three classes of inductively monothetic groups of Theorem 5.15 and observe the following: (i) Any nondegenerate quotient of a group being a one-dimensional compact connected abelian group has the same properties. (ii) Any quotient of a group that is standard inductively monothetic locally compact has the same properties, since passing to the quotient preserves primary components. (iii) Any nondegenerate proper quotient of the discrete group ℚ is a quotient of the discrete group ℚ/ℤ and is therefore is a discrete Π-procyclic group. Therefore we have the following. Corollary 5.20. For a locally compact inductively monothetic group G the following conditions are equivalent: (1) G is compact, (2) G is monothetic and not isomorphic to ℤ. Proof. Assume (1). We inspect the possible cases of Theorem 5.15. If G is a onedimensional compact connected group, then its character group is isomorphic to a subgroup of ℚ and thus G is monothetic. (See, e. g., [54, Example 8.75].) Now assume that G is totally disconnected. By condition (1) it cannot be a nonsingleton subgroup of ℚ. Then by Theorem 5.15 (2) G is isomorphic to a product of compact p-groups Mp which by Definition 5.9 are procyclic. Hence G is procyclic, that is, monothetic. Condition (2) implies (1) by Weil’s lemma [54, Proposition 7.43]. Corollary 5.21. Let G be a locally compact abelian torsion group. Then every inductively monothetic subgroup is a discrete subgroup of G isomorphic to a subgroup of ℚ/ℤ. Proof. Let H be an inductively monothetic subgroup of the torsion group G = ∏ Gp ⊕ ⨁ Gp p∈ϕ

p∈π\ϕ

5.1 Classifying inductively monothetic subgroups | 135

by Proposition 3.11. Then H = ∏ Hp ⊕ ⨁ Hp p∈ϕ

p∈π\ϕ

by Theorem 5.15 is a standard inductively monothetic group according to Definition 5.9. Since Gp is a torsion group, Hp is either isomorphic to ℤ(p∞ ) or cyclic according to the list of Definition 5.9. Thus H is a locally compact subgroup of G algebraically isomorphic to a subgroup of ⨁p∈π ℤ(p∞ ) = ℚ/ℤ. A countable locally compact topological group is discrete by the Baire category theorem. Corollary 5.22. A quotient of an inductively monothetic group is inductively monothetic. Recall the concept of a standard inductively monothetic group from Definition 5.9 (ii). Corollary 5.23. For a locally compact abelian group G consider the following statements: (1) G is inductively monothetic, (2) every compact subgroup of G is monothetic, (3) every compact open subgroup of G is monothetic. Then (1)⇒(2)⇒(3), and if G is compactly ruled, then both conditions (1) and (2) are equivalent. If, in addition, G0 is compact, then (3)⇒(1). Proof. The implication (1)⇒(2) follows from Corollary 5.20 and certainly (2)⇒(3). Now assume that G is compactly ruled and assume (2). Then Proposition 1.3 says that every finite subset of G is contained in a compact subgroup C; by hypothesis (2), C is monothetic. Thus (1) holds. Now assume that, moreover, G0 is compact. Then there exists a compact open subgroup U, and then every compact subgroup C is contained in the compact open subgroup C + U which is monothetic by (3), whence C is monothetic. Thus (3) implies (2) and hence (1). We recall from Corollary 1.12 that a solvable locally compact group G is compactly ruled if and only if G = comp(G). If G is known to be inductively monothetic, then this means that G is not discrete and isomorphic to a subgroup of ℚ in view of Corollary 5.16. Examples of groups from the classification Theorem 5.15 are given by each of the groups ℚp or by local products of the type described in the following. Example 5.24. Define G = ∏loc p (ℤp , pℤp ). Then C = ∏p pℤp is an open subgroup of ̃ G isomorphic to ℤ = ∏p ℤp , while its factor group G/C is isomorphic to the infinite discrete group ⨁p ℤ(p). The inclusion map induces an injective continuous morphism ̃ = ∏ ℤp with dense image and is not open onto its image. ε: G → ℤ p

136 | 5 Inductively monothetic groups Examples of this kind illustrate the considerable variety of standard inductively monothetic groups in Theorem 5.15. We recall from this theorem and Definition 5.9 that any Π-procyclic inductively monothetic group H may be identified with a local product ∏loc p (Hp , Cp ), where ∏p Cp is a compact open subgroup and where Hp are the p-Sylow subgroups of H each of which is a procyclic p-group, i. e., either cyclic or isomorphic to ℤp . Any closed subgroup of

K ≤ H with p-Sylow subgroups Kp then may be written ∏loc p (Kp , Cp ∩ Kp ). The index p ranges through the set π of all primes. In particular, H/K ≅ ∏loc p (Hp /Kp , (Cp + Kp )/Kp ). This requires a bit of checking: Let (hp + Kp )p in the right-hand side be a π-tuple such that for almost all p we have hp = cp + kp for cp ∈ Cp and kp ∈ Kp ; then for almost all p we have kp = cp󸀠 ∈ Cp whence

hp = cp + cp󸀠 ∈ Cp for almost all p and so (hp )p ∈ ∏loc p (Hp , Cp ) = H. Therefore, H/K is compact if and only if ⨁p (Hp /(Cp + Kp )) is finite. Therefore we have the following observation. Lemma 5.25. Let H be a Π-procyclic inductively monothetic group and K a closed subgroup. Then K is cocompact if and only if the set of primes p such that Hp ≠ Cp + Kp is finite. The following theorem will be crucial for the later developments. Theorem 5.26 (The gamma theorem). Let Γ be an inductively monothetic locally compact group acting faithfully on a locally compact abelian group G as a group of scalar automorphisms and assume that Γ contains at least one element acting neither as idG nor as −idG . Then G is periodic and Γ is either isomorphic to (1) a nontrivial discrete proper subgroup of ℚ or (2) a Π-procyclic group.

Proof. Let η denote the canonical morphism from Γ to SAut(G) that is guaranteed by the hypotheses. Since the action is faithful, η is injective. By Theorem 4.24, if G is not periodic, then some element of Γ acts trivially on G contrary to the hypothesis. Hence G is periodic. So by Theorem 4.25 SAut(G) is a profinite group. Since a profinite group cannot contain any nontrivial divisible subgroups and since η is injective, Γ does not contain any divisible subgroup. We apply Theorem 5.15 and arrive at the conclusion. Note that G is the union of its compact (and therefore Γ-invariant) subgroups. The case that G is a compact p-group and Γ is a monothetic p-group was extensively discussed in [57]. The complementary situation to that of the gamma Theorem 5.26 is that of a locally compact abelian group G and the group Γ2 := {idG , −idG } ⊆ SAut(G) acting faithfully on it.

5.2 The subspace of inductively monothetic subgroups | 137

Remark 5.27. The action of Γ2 amounts to the action of the involution τ = −idG discussed in Remark 4.10.

5.2 The subspace of inductively monothetic subgroups In a compact totally disconnected group G, the space of monothetic subgroups is closed in 𝒮𝒰ℬ(G) (see [57]). We shall have to consider limits of nets of inductively monothetic subgroups of a locally compact group. Therefore the following observation will be relevant. Proposition 5.28. Let G be any totally disconnected compactly ruled group. Then the set of inductively monothetic subgroups is closed. Proof. Since G is compactly ruled, it is the directed union of its compact open subgroups. A subgroup H ∈ 𝒮𝒰ℬ(G) is in the closure of the set of inductively monothetic groups in 𝒮𝒰ℬ(G) iff there is a net (Hj )j∈J of inductively monothetic groups converging to H. By Corollary 1.23 H = limj Hj if and only if, for every open compact subgroup U, we have H ∩ U = limj (Hj ∩ U). However, since G is totally disconnected, so is each of the compact groups U. In the compact totally disconnected group U, the subspace of monothetic groups in 𝒮𝒰ℬ(U) is closed. Therefore H ∩ U is an open monothetic subgroup of H. Thus H is the union of compact open monothetic subgroups and hence it is inductively monothetic (see Notation 5.1). In [57], Example 3.3, an example of a connected compact group C is given in which the set of monothetic subgroups is not closed in 𝒮𝒰ℬ(C). The group G = ℚp is an inductively monothetic group which is itself not Π-procyclic but is such that the set of Π-procyclic subgroups of 𝒮𝒰ℬ(G) is dense in 𝒮𝒰ℬ(G). Thus in Proposition 5.28 the words “inductively monothetic subgroups” cannot be replaced by “Π–procyclic subgroups”, let alone by the words “procyclic subgroups”. Proposition 5.28 will apply to periodic near abelian groups later. Proposition 5.29. Let G be a locally compact group and 𝒩 a filter basis of closed normal subgroups N such that G/N is inductively monothetic and G = lim G/N. Then G is in←󳨀󳨀𝒩 ductively monothetic. Proof. We shall repeatedly apply the classification Theorem 5.15. Case (i). G is connected. Then all G/N are connected and then are one-dimensional ̂ is the union of a directed set of discrete subgroups monothetic. By duality, then G ̂ is discrete torsion-free of rank 1, and thus which are torsion-free of rank 1; therefore G G is compact connected one-dimensional (and monothetic). Thus we need to consider only the case that the quotients G/N are totally disconnected cofinally, and thus G is totally disconnected.

138 | 5 Inductively monothetic groups Case (ii). One of the homomorphic images G/N is isomorphic to a subgroup of the discrete group ℚ. Let M ∈ 𝒩 , M ⊆ N. Then G/M is an inductively monothetic group and has as a quotient group a nondegenerate subgroup of the discrete group ℚ. An inspection of the possibilities (1) and (2) of the classification Theorem 5.15 shows that G/M itself must be isomorphic to a subgroup of ℚ. We claim that M = N. Indeed suppose M ≠ N. Then G/N is a proper homomorphic image of G/M. The fact that G/M is a nondegenerate subgroup of ℚ then implies that G/N is a torsion group, which is impossible since G/N is a nondegenerate subgroup of ℚ as well. That is, M = N. The assumption that 𝒩 converges to 1 then implies that N = {1}. We are left with the case that all quotients G/N are of type (2) of the classification Theorem 5.15. Case (iii). We now assume that all quotients G/N are periodic and inductively monothetic. Then G is periodic, and in order to prove that G is inductively monothetic, we show that each p-Sylow subgroup has p-rank ≤ 1. We observe that (G/N)p ≅ Gp /Np by Braconnier’s Theorem 3.3. Since G/N is inductively monothetic we know that the p-rank of (G/N)p is ≤ 1. So rankp Gp /Np ≤ 1 for all primes p. The canonical projection prp : G → Gp of Braconnier’s theorem maps N to Np and the limit lim𝒩 G/N can be calculated componentwise, whence Gp ≅ (lim𝒩 (G/N))p ≅ lim𝒩 (G/N)p , and so Gp ≅ lim𝒩 Gp /Np . By Theorem 5.15 Gp /Np is cofinally isomorphic to ℚp , ℤ(p∞ ), or {0}, or else it is p-procyclic. This implies that Gp itself is of one of these types, which is our claim. We note that some case distinction in the proof is necessary since a projective limit of compact connected monothetic groups may not be monothetic in general, as was noted in [57], Example 3.3.

5.3 On divisibility and ℤp The following proposition is added to show which examples are allowed typically by Theorem 5.26. Recall ℤp ⊆ ℚp and ℚ ⊆ ℚp . We define 𝔽p := ℤ[ p1 ] to be the subring of the rationals consisting of fractions with numerator a power of p, and we let p 𝔽 denote the subring of the rationals consisting of fractions ba , where b is coprime to p. We note that, algebraically, 𝔽p is the ring adjunction of p1 to ℤ and that p 𝔽 is the p󸀠 -localization of ℤ. Proposition 5.30. (i) The additive group ℤp does not contain any nontrivial divisible subgroups, (ii) ℚ ∩ ℤp = p 𝔽, (iii) ℚ + ℤp = ℚp , (iv) ℤp /p 𝔽 ≅ ℝ (as abstract groups),

(v) ℤp /ℤ ≅ ℝ ×

p𝔽

ℤ

, where

p𝔽

ℤ

≅ ⨁p=q̸ ℤ(q∞ ); in particular, ℤp /ℤ is divisible.

5.4 The extensions of a locally compact group by a Π-procyclic group | 139

Proof. (i) The group of p-adic integers ℤp is not the direct sum of two proper subgroups (see [32], p. 170 ff.). A divisible subgroup of an abelian group is a direct summand (see, e. g., [54] Corollary A1.36 (i)). So statement (i) follows. (ii) A number q ∈ ℚp is of the form q = pn z with a unique smallest n ∈ ℤ and a unique unit z ∈ ℤp . We have q ∈ ℤp iff n ≥ 0. A rational number is of the form ±pn z with a unique smallest n ∈ ℤ and a unique positive rational number z relatively prime to p. Thus q ∈ ℚ ∩ ℤp iff n ≥ 0 and z ∈ ℚ relatively prime to p iff q ∈ p 𝔽. It also follows from this that 𝔽p ∩ ℤp = 𝔽p ∩ p 𝔽 = ℤ. (iii) The sum 𝔽p + ℤp is an additive subgroup of ℚp containing ℤp and such that ℤ(p∞ ) ≅ ℚp /ℤp ⊇ (𝔽p + ℤp )/ℤp ≅ 𝔽p /(𝔽p ∩ ℤp ) = 𝔽p /ℤ = ℤ(p∞ ). Since ℤ(p∞ ) has no proper infinite subgroups, ℚp /ℤp = (𝔽p + ℤp )/ℤp follows, and this implies 𝔽p + ℤp = ℚp . A fortiori, ℚ + ℤp = ℚp . ℤp ℤp ∩ℚ

(iv) Using (ii) we compute ℤp /p 𝔽 =

≅

ℚ+ℤp ℚ

= ℚp /ℚ in view of (iii). Now ℚp

is a rational vector space of dimension c (the cardinality of the continuum) and ℚ is a one-dimensional vector subspace. Thus ℚp /ℚ is a rational vector space of dimension c and thus is isomorphic to ℝ as a rational vector space. ℤp /ℤ ℤ (v) Firstly, we note that 𝔽/ℤ ≅ 𝔽p ≅ ℝ. Secondly, we observe that p 𝔽/ℤ = p

p

⨁p=q̸ ℤ(q∞ ). Since this group is divisible and each divisible subgroup is a direct

summand (see [54], A1.36 (i)) we conclude

ℤp ℤ

≅ℝ×

p𝔽

ℤ

as asserted.

If we write the circle group additively as 𝕋 = ℝ/ℤ ≅ ℝ × ℤp ℤ

ℚ , ℤ

then we notice that

ℤp /ℤ ≅ and 𝕋 ≅ × ℤ(p ). As known (see, e. g., [93, Theorem 4.4.7]), the compact ring ℤp contains a copy of ℤp in its multiplicative group of units whence SAut(G) contains a copy of ℤp and there𝕋 ℤ(p∞ )

∞

fore any discrete subgroup of ℤp , such as p 𝔽, or the noncompact group ∏loc p (ℤp , pℤp ) of Example 5.24 can appear as Γ for G = ℤp in Theorem 5.26.

5.4 The extensions of a locally compact group by a Π-procyclic group From Definition 1.7 we recall that a locally compact group G is called an extension of a closed normal subgroup A by a group P if the quotient group G/A is isomorphic to P. We are interested in the case that P is an inductively monothetic group (see Notation 5.1). We further saw in the classification theorem of inductively monothetic groups, Theorem 5.15, that a typical class of these groups are the so-called Π-procyclic ones (see Definition 5.9 ff.). Therefore we now study extensions of an arbitrary locally compact group A by a Π-procyclic group P. We are particularly interested in finding out when it is possible, in such an extension, to find a closed (and hopefully also Π-procyclic) subgroup H of G such that in fact we have a product representation G = AH.

(#)

140 | 5 Inductively monothetic groups In Example 11.13 we see that even in the case that G is a countable subgroup of the abelian group ℝ/ℤ this is not always possible; this example also serves other purposes and therefore is more complex than would be required for the present purposes. So we immediately record a more primitive but typical example as a cautionary note. Example 5.31. Let G = ℤ and A = 3ℤ ≅ ℤ. Then G is the extension of A by a cyclic group of prime order and G is as far from having a product decomposition (#) with a Π-procyclic factor H as is conceivable. On the other hand, the group H = 2ℤ satisfies G = AH (and A ∩ H = 6ℤ) where H is a cyclic group. Since the significance of the factor H is clear from Proposition 1.31 we aim to prove in the remainder of this section the following result on locally compact groups, which represents a first portion of our efforts of establishing supplementation theorems for abelian extensions; a second will emerge in Theorem 7.36. Theorem 5.32 (Main theorem on a product representation (#)). Let G be a locally compact group with a compact normal subgroup A. Then G has a product representation (#) if at least one of the following conditions is satisfied: (i) G/A is Π-procyclic, (ii) the subgroup A is open and G/A is a rank 1 torsion-free abelian group. Moreover, in case (ii) the product AH is semidirect, and in case (i) the closed subgroup H is Π-procyclic. The simplest occurrence of Theorem 5.32 is that G/A is monothetic for which we do not even need the full hypothesis that A be compact. We recall that comp(G) is the union of all compact subgroups of G (see Section 1.1) and that there are two kinds of monothetic subgroups of locally compact groups by Weil’s lemma (see, e. g., [54], Proposition 7.43): (a) G/A is infinite cyclic, (b) G/A is compact. Proposition 5.33. Let G be a locally compact group and A a closed normal subgroup such that G/A is monothetic. Then the following two conditions are equivalent: (1) there is a compact monothetic subgroup H = ⟨b⟩ in G such that G = AH as in (#), (2) A comp(G) = G. If G/A is infinite cyclic, then H ≅ ℤ and G is the semidirect product A ⋊ H. Proof. The implication (1)⇒(2) is trivial. We now assume (2). Since G/A is monothetic there is an element b ∈ G such that Ab is a generator of G/A. By (2) we may select b in comp(G). Then H := ⟨b⟩ is a compact monothetic subgroup. Now p: H → G/A, p(h) = Ah is a surjective morphism of compact groups and so G = AH.

5.4 The extensions of a locally compact group by a Π-procyclic group

| 141

Now assume that G/A is infinite cyclic. Let b ∈ G be an element such that Ab is one of the two generators of G/A and set H = ⟨b⟩. Then n 󳨃→ Abn = (Ab)n : ℤ → G/A is an isomorphism of discrete topological groups since A is open, G/A is infinite cyclic, and Ab is a generator of G/A. Its kernel A ∩ H is therefore singleton, and G = AH. The assertion follows. Corollary 5.34. Let G be a locally compact group satisfying G = comp(G) and A a closed normal subgroup such that G/A is monothetic. Then there is a compact monothetic subgroup H = ⟨b⟩ in G such that G = AH as in (#). What Proposition 5.33 and its corollary leave to be desired is illustrated in the following example. Example 5.35. Let G = ℤ × ℤp (p > 2) and A = ℤ × {0}. Then G is an abelian and thus near abelian group with base A such that G/A ≅ ℤp is monothetic. We write ℤ ⊆ ℤp . Now we let b = (1, 1) ∈ ℤ × ℤp and set H = ⟨b⟩. Then as in Proposition 5.33 the element Ab = ℤ × {1} is a generator of G/A. However, H = {(n, n) : n ∈ ℤ} ≅ ℤ is a closed monothetic subgroup of G and AH = ℤ × ℤ, which is a dense proper subgroup of G. Thus AH ≠ G. We have comp(G) = {1} × ℤp so that G = A comp(G) and condition (2) of Proposition 5.33 is satisfied. It is the “wrong” choice of the generator b which fails to produce the desired compact monothetic supplement H. The search for a supplementary subgroup H in a locally compact group G with a closed normal subgroup A in the case that G/A is monothetic (e. g., procyclic) was comparatively simple. The next complication along this line is that G/A is a Π-procyclic group. From Definition 5.9 through Proposition 5.14 we shall recall some notation and information concerning Π-procyclic groups. Let π = {2, 3, 5, . . . } denote the set of prime numbers in their natural order and let P = ∏loc n∈ℕ (Ppn , Cpn ) be a Π-procyclic group. That is, while n ranges through ℕ, Ppn is a monothetic pn -group (that is, a pn -procyclic group) and Cpn is an open subgroup. See Definition 2.49. For a natural number m we define P[m] = {(xpk )k∈ℕ ∈ P : (∀n ≥ m) xpn ∈ Cpn }. Definition 5.36. We shall call the monothetic subgroups P[m] the standard monothetic subgroups of P. We noted that P[1] = C := ∏p Cp and that each P[m] is a compact open monothetic subgroup of P with a generator gm = g1 ⋅ ⋅ ⋅ gm−1 so that P[m] = ⟨gm ⟩. Moreover, if we set θ(m) := |P[m]/C| = ∏k 2 { } ≅ (ℤp , +) k = 2, if p = 2

of the ring (ℤp , +, ⋅) acts by scalar automorphisms on the direct product ∏n∈ℕ Gn (see

Corollary 4.1). Therefore we can form the local product A := ∏loc n∈ℕ (Gn , Cn ) (see Theorem 3.3) and get H to act on A by scalar automorphisms (see Corollary 4.11). Therefore we obtain a periodic near abelian p-group G := A ⋊ H with base group A and compact scaling group H; the base group A and thus the group G is noncompact in both case A and case B. In the preceding examples, the base group A was (potentially) noncompact while the scaling group was compact. The following example of a locally compact group illustrates the reverse situation. Remark 11.5. Example 11.4 produces a near abelian p-group in which a base group A splits as a normal semidirect factor with a compact complement H ≅ ℤp ; we encountered such groups in Theorem 5.41. If, in the same example, we pass to a subgroup of H that is abstractly isomorphic to ℤ and equip G with a topology that declares A to be an open subgroup, then the new example resulting in this fashion gives an example of a class of near abelian groups which we discuss in Proposition 5.33. Example 11.6. We let A = ∏p ℤp with p ranging through all prime numbers. Let Hp := (1 + pk ℤp , ×),

k = 1, if p > 2 { } ≅ (ℤp , +). k = 2, if p = 2

Then each Hp acts by scalar automorphisms on ℤp , and so H := ∏p Hp acts by scalar automorphisms on A = ∏p ℤp , componentwise. Then A ⋊ H ≅ ∏(ℤp ⋊ Hp ) p

11 A list of examples |

221

is a compact near abelian group. If Kp = (1 + pk+1 ℤp , ×), then Hp /Kp = Cp is cyclic of

order p, L := ∏loc p (Hp , Kp ) is a noncompact inductively monothetic group (see Example 5.24), and ε: L → H induced by the inclusion is an injective continuous morphism with dense image. Accordingly, G = A ⋊ L is a noncompact near abelian group with a compact base group ≅ A and a noncompact scaling group ≅ L which is mapped injectively and continuously into the compact near abelian group A ⋊ H with dense image by the morphism idA × ε. The following is an example of a locally compact abelian group G with a base group A for which G/A is an infinite torsion group. Example 11.7. Choose primes pi and qi such that pi ≡ 1 (mod qi ). Then there is ri ∈ ℤqi p such that ri i = 1. Set A := ∏i∈ℕ ℤqi and let a generator bi of ℤ(pi ) act trivially on ℤqj if

j ≠ i and as z bi := z ri . Forming the discrete direct sum H := ⨁i∈ℕ ℤ(pi ) we can extend the actions of the bi to a scalar action of H. The semidirect product G := A ⋊ H has a torsion scaling group. Next we provide an example with G/A torsion-free. Example 11.8. Let p be an odd prime and recall from Proposition 4.17 that SAut(ℤp ) ≅ ℤp × ℤ(p − 1). Let H = p 𝔽× denote the set of all rational numbers ba with b relatively prime to p. By Proposition 5.30 (ii) every element in H is a unit in the ring ℤp . Put A := ℤp and equip H with the discrete topology. Then G := A ⋊ H has a torsion-free discrete countable scaling group. Example 11.9. Consider the Prüfer group ℤ(2∞ ) as a discrete group and the profinite group of 2-adic integers ℤ2 . We let the multiplicative group 1 + 4ℤ2 of 2-adic integers act by scalar multiplication on ℤ(2∞ ). Namely, the topological generator g = 1 + 2n of 1 + 2n ℤ2 acts by multiplication by g on ℤ(2∞ ). Then the semidirect product G := ℤ(2∞ ) ⋊ (1 + 4ℤ2 ) is near abelian. Since ℤ(2∞ ) is discrete the family of sets 1 × 2n ℤ2 , n = 2, 3, . . . , forms a basis of identity neighborhoods of 1. Now (a, 1)(0, g)(−a, 1) = (a, g)(−a, 1) = (a − (1 + 2n )⋅a, g) = (−2n ⋅a, g), and thus the conjugacy class of g contains ℤ(2∞ ) × {g}. Since any identity neighborhood U, no matter how small, but contained in {0} × 4ℤ2 , contains an element g = 1 + 2n for sufficiently large n whose conjugacy class ℤ(2∞ ) × {g} is not contained in {0} × (1 + 4ℤ2 ), the neighborhood U cannot be an invariant identity neighborhood. Hence G does not contain arbitrarily small invariant identity neighborhoods. In particular, G does not contain arbitrarily small compact open normal subgroups and thus cannot be prodiscrete.

222 | 11 A list of examples Slightly generalizing this example one may consider the discrete group A := ̃ = ∏ ℤp act. Thus G := ℚ/ℤ ⋊ ℤ ̃ is a totally disconℚ/ℤ ≅ ⨁p ℤ(p∞ ) and let H := ℤ p nected near abelian group which is not a strict projective limit of discrete groups, i. e., is not prodiscrete. Example 11.10. Consider the holomorph G := ℚp ⋊ ⟨α⟩ for α the automorphism that sends any r ∈ ℚp to r/p. Then G has no compact open normal subgroup whatsoever. The conjugacy classes of (r, αn ) for n ≠ 0 fail to be compact. The following example of a compact group contains A on which G acts by scalars and G/A is monothetic but not inductively monothetic. So G is not near abelian; but the example shows that connectivity can be more tricky if, in the definition of a near abelian group, one begins to vary the hypothesis on G/A. Example 11.11. Let ℤ(9) be the natural (1+3ℤ(9), ×) ≅ ℤ(3)-module and let Γ = ℤ(3)×𝕋 act on ℤ(9) by the scalar action of the first factor. Form the semidirect product ℤ(9) ⋊ (ℤ(3)×𝕋). The subgroup 3ℤ(9)×{0}× 31 ℤ/ℤ is central and contains a diagonal subgroup D generated by (3 + 3ℤ(9), 0, −1/3 + ℤ). Then G := (ℤ(9) ⋊ (ℤ(3) × 𝕋))/D has a finite base group of order 9 and a circle group as nontrivial identity component which intersects the base group in a subgroup of order 3 and which does not split over the base group. Note that Γ is monothetic but not inductively monothetic. Example 11.12. Let α󸀠 : ℤ2 → {1, −1} be the morphism with kernel 2ℤ2 and let α: ℤ2 → SAut(ℤ2 ) be given by α(z)(m, n) = α󸀠 (z)⋅(m, n). Set G = ℤ2 ⋊α ℤ2 . This is an A-trivial near abelian locally compact group for the base group A = ℤ2 × {0}. Then CG (A) = ℤ2 × 2ℤ2 is an abelian closed subgroup of index 2 in G. In order to create an injective morphism μ: ℤ2 → 2ℤ2 let λ ∈ ℤ2 \ 2 𝔽 according to Proposition 5.30 (ii); then μ(m, n) = 2(m + λn) will do. So L := graph(μ) = {((m, n), 2(m + λn)) ∈ G : (m, n) ∈ ℤ2 } is a discrete and hence closed abelian subgroup L of G with L ∩ A = {0}. Accordingly, L/(L ∩ A) ≅ ℤ2 fails to be inductively monothetic and thus L ∩ A fails to be a base for L. Notice that the subgroup LA = ℤ2 × (2ℤ + 2λℤ) fails to be locally compact and is in fact dense in CG (A). The natural bijective morphism L → LA/A is not an isomorphism. The following example is relevant for the question of the existence of scaling groups. Example 11.13. There exists an abelian group G with the following properties: (i) The torsion subgroup A := tor G is isomorphic to ⨁ ℤ(p).

p=2,3,5,...

11 A list of examples |

223

(ii) The torsion-free quotient G/A is isomorphic to 1 ⋅ℤ ⊆ ℚ. 2⋅3⋅5 ⋅⋅⋅p p=2,3,5,... ⋃

(iii) The torsion subgroup A is not a direct summand. The group G is a subgroup of ℝ/ℤ. (This is a recent construction due to D. Maier [73, 74].) This example is an indication that there is no chance of proving a general existence theorem for a scaling group H for a near abelian group G with given base A.

|

Part III: Applications

Introduction In this chapter we apply the investigations of near abelian groups in order to recover and augment results found by Yu. Mukhin. The subsequent sections are devoted to this task. By definition a locally compact group G is topologically quasihamiltonian if for any closed subgroups X and Y the set XY is a closed subgroup of G (see [68]). For similar investigations of algebraic and Liegroups see [25]. The description of topologically hamiltonian groups is due to S. P. Strunkov (see [102]) and again by P. Diaconis and M. Shahshahani (see [26]). A classification of topologically quasihamiltonian groups has been given by Mukhin in [82] and we re-prove and augment his results in Chapter 12. The main results comprise Theorem 12.5 (the p-group case), Theorem 12.11 (the periodic group case), and Theorem 12.17 classifying the nonperiodic topologically quasihamiltonian groups. A locally compact group G is topologically modular provided its lattice of closed subgroups satisfies Dedekind’s law. A classification of topologically modular groups has been given by Mukhin (see [80, 81] and is presented in Chapter 13. As in [81], for periodic groups, we shall restrict ourselves to compactly ruled groups. The main results are Proposition 14.24 (showing that the class of topologically modular p-groups agrees with the class of topologically quasihamiltonian p-groups) and Theorem 13.31 (describing compactly ruled topologically modular groups). Finally it turns out that nonperiodic topologically modular groups are always topologically quasihamiltonian; see the discussion in Section 13.6. However, Example 12.16 shows that the converse need not be true. A locally compact group G is strongly topologically quasihamiltonian provided that for any closed subgroups X and Y the set XY is a closed subgroup. A classification of strongly topologically quasihamiltonian groups will be given in Chapter 14. We first complete Mukhin’s classification of the abelian strongly topologically quasihamiltonian groups (see Theorem 14.22). The complete classification of p-groups has been given by Mukhin in [78]. The main results are Theorem 14.28 together with Proposition 14.31 (the periodic case) and Theorem 14.39 (the nonperiodic case). The idea behind the classification is here to use the fact that any strongly topologically quasihamiltonian group is near abelian and then to give necessary and sufficient conditions on the base subgroup. Let us summarize the relations between the various classes.

228 | Introduction

The periodic case Theorem 12.11 provides a description of all periodic topologically quasihamiltonian groups. The subsequent comments reveal that every periodic topologically quasihamiltonian group is near abelian and hence compactly ruled. – Not every topologically quasihamiltonian group is topologically modular; see Example 13.2. – Not every topologically modular group is topologically quasihamiltonian; see Corollary 13.19. – Every strongly topologically quasihamiltonian group is both topologically quasihamiltonian and topologically modular. – A locally compact compactly ruled p-group is topologically modular iff it is strongly topologically quasihamiltonian. (See Proposition 14.24.) – There are groups that are both topologically quasihamiltonian and topologically modular, and yet they are not strongly topologically quasihamiltonian; see Remark 14.6(c).

The nonperiodic case Here we shall talk about groups containing a noncompact element. – Every topologically modular group is topologically quasihamiltonian; see Subsection 13.6. The converse is false, as shown by Example 12.16. – A nonperiodic locally compact group is strongly topologically quasihamiltonian if and only if it is both topologically quasihamiltonian and topologically modular; see Theorem 14.40.

12 Classifying topologically quasihamiltonian groups Our first application of the results on near abelian groups consists of completing the classification of locally compact quasihamiltonian groups, initiated by F. Kümmich in [68, 69] and completed by Yu. Mukhin in [82]. In particular he proved there that a nonabelian topologically quasihamiltonian group G must be totally disconnected. When G has only compact elements, i. e., each element is contained in a compact subgroup of G it will turn out that G is periodic, is near abelian, is the local product of topologically quasihamiltonian p-groups; see Theorem 12.11 and the comments thereafter. The p-factors themselves satisfy relations resembling those in Iwasawa’s classification of (locally) finite quasihamiltonian p-groups. See Theorem 12.5. Finally we turn to the situation when G contains a discrete subgroup isomorphic to ℤ. Then, similarly as in the discrete case, already handled by Iwasawa, G turns out to be near abelian and the compact elements form a base group. The full description of such a group appears in Theorem 12.17 and indeed resembles the one in the discrete situation.

12.1 Generalities According to F. Kümmich, see [68], a locally compact group G is topologically quasihamiltonian provided XY = YX holds for any pair X, Y of closed subgroups of G, that is, if and only if XY is a closed subgroup of G. Let us first collect some information about such groups from the cited reference. Proposition 12.1. The class of locally compact topologically quasihamiltonian groups is closed under forming subgroups, continuous homomorphic images, and projective limits with open bonding maps and compact kernels. Moreover, a nonabelian locally compact topologically quasihamiltonian group is zero-dimensional. Proof. This is the content of Hilfsatz 2, 3, and 4 in [68]. Recording a standard result from topology, we have the following. Lemma 12.2. Let (Ai )i be a family of subsets of a topological space X. Then ⋃ Ai = ⋃ Ai . i

i

Proof. The containment “⊇” is clear. Conversely, let C denote the closed subset ⋃i Ai .

Then for all i ∈ I we have Ai ⊆ C, whence ⋃i∈I Aj ⊆ C. Thus ⋃i∈I Aj ⊆ C = C and so we have “⊆” as well. The next result compares with [57, Proposition 2.7]. During its proof we shall make use of Lemma 12.2. https://doi.org/10.1515/9783110599190-012

230 | 12 Classifying topologically quasihamiltonian groups Proposition 12.3. Let G be a locally compact group and assume that XY = YX holds for all monothetic subgroups X and Y of G. Then G is topologically quasihamiltonian. Proof. If X is a closed subgroup of any topological group, then X = ⋃x∈X ⟨x⟩. Fix closed subgroups X and Y. Note first that for x ∈ X and y ∈ Y the assumptions yield the identity of subgroups ⟨x⟩⟨y⟩ = ⟨x, y⟩ = ⟨y⟩⟨x⟩. Making use of this and Lemma 12.2 one finds XY = ( ⋃ ⟨x⟩)( ⋃ ⟨y⟩) = x∈X

=

⋃

y∈Y,x∈X

y∈Y

⟨y⟩⟨x⟩ =

⋃

x∈X,y∈Y

⟨x⟩ ⟨y⟩ ⊆

⋃

x∈X,y∈Y

⟨x, y⟩

⋃ ⟨y⟩⟨x⟩ ⊆ ( ⋃ ⟨y⟩)( ⋃ ⟨x⟩) = YX. y∈Y

y∈Y,x∈X

x∈X

Changing the roles of X and Y yields the reverse containment YX ⊆ XY. Hence G is topologically quasihamiltonian. The following observation fits here. Proposition 12.4. Let G be a topologically quasihamiltonian group and X a closed periodic sigma-compact subgroup. Then XY = YX holds for any closed sigma-compact subgroup Y of G. Proof. The sigma-compact subgroup Y can be presented as an ascending union of compact subgroups Yk , i. e., ∞

Y = ⋃ Yk . k=1

Since G, by assumption, is topologically quasihamiltonian and for any k in ℕ the subgroup Yk is compact, one deduces XYk = XYk = Yk X = Yk X. Therefore the chain of equalities ∞

∞

∞

∞

k=1

k=1

k=1

k=1

XY = X( ⋃ Yk ) = ⋃ XYk = ⋃ Yk X = ( ⋃ Yk )X = YX completes the proof.

12.2 The p-group case

| 231

12.2 The p-group case The first step towards a classification of topologically quasihamiltonian locally compact groups is to provide a classification of topologically quasihamiltonian locally compact p-groups—generalizing at the same time the classification theorem for quasihamiltonian compact p-groups in [57, Theorem 7.2] and Iwasawa’s structure theorem for locally finite quasihamiltonian p-groups (cf. [97, Theorem 2.4.14]). This result has also been proved by Yu. Mukhin in [81] with the use of different methods. Our approach is based on some prior experience with the classification of compact topologically quasihamiltonian groups in [57] and our present treatment of trivial locally compact near abelian groups in Chapter 8. Since here we are concerned with locally compact p-groups, our situation now is simpler than the one treated in Chapter 8: What was called the supplementary part G+ in Definition 8.3 is simply the base group A, while, for our consideration of the prime p = 2, the p-group G is exactly what was called there the effective part G[2] . We recall from Definition 8.5 the definition of the generalized quaternion groups Mn for n = 2, 3, 4, . . . . In fact we describe several equivalent conditions, and n = 2 yields the familiar eight-element quaternion group. We have |Mn | = 2n+1 . The next Theorem is from [82]. Theorem 12.5. A locally compact p-group G is topologically quasihamiltonian if and only if G is near abelian with a base group A and an inductively monothetic p-group G/A and at least one of the following statements holds: (a) G is abelian, (b) there is a p-procyclic scaling group H = ⟨b⟩ such that G = AH and there is a natural s number s ≥ 1, respectively, s ≥ 2, if p = 2, such that ab = a1+p for all a in A; the group G is A-nontrivial, (c) p = 2 and G ≅ A2 × Mn , where A2 is an exponent 2 locally compact abelian group according to Remark 8.7, and Mn is the generalized quaternion group of order 2n+1 ; in this case, A = A2 × ⟨a⟩ ≅ ℤ(2)(I1 ) × ℤ(2)I2 × ℤ(4) with a as in Definition 8.5 for suitable sets I1 and I2 . The group G is A-trivial. Remark 12.6. In case (a), Example 7.42 shows that a discrete abelian p-group exists with a subgroup A such that G/A ≅ ℤ(p∞ ) such that there is no subgroup H with G = A + H (in additive notation). Hence a scaling subgroup may not exist. Proof. We may assume right away that G is not abelian. Now let G be a locally compact topologically quasihamiltonian group. We need to show that (b) or (c) holds. By the definition of a topologically quasihamiltonian group G the product of two compact subgroups is a compact group. Thus comp(G) is compactly ruled. In particu-

232 | 12 Classifying topologically quasihamiltonian groups lar, in light of Proposition 1.3, it turns out that a locally compact topologically quasihamiltonian p-group G is compactly ruled. By [57], Theorem 7.1 and the arguments in the proof of Theorem 7.2, the present theorem is true for all compact groups. Thus all compact open subgroups, being topologically quasihamiltonian, are near abelian. By Proposition 9.11 it follows that G is near abelian. So there is a locally compact abelian subgroup A as base with a locally compact periodic inductively monothetic p-group as quotient group G/A. Thus either G/A is divisible, namely, ℚp or ℤ(p∞ ) or G/A is a procyclic p-group. If G/A is divisible, then G/A = CG (A)/A by Theorem 6.4 (ii), and so G = CG (A), which means that G = CG (A) is abelian by Theorem 6.4 (i); but this case was explicitly ruled out. Therefore G/A is procyclic and by Proposition 5.33 there is a scaling subgroup H = ⟨b⟩ for some generator b ∈ H such that G = AH. Since b acts via inner automorphism on A by scalar multiplication, by the arguments of the proof of Theorem 7.2 in [57], a natural number s ≥ 1 for odd p and s ≥ 2 if p = 2 may be found such that (possibly after a replacement of s the generator b by an equivalent one), for all a ∈ A, we have ab = a1+p while ab = a−1 does not hold for all a ∈ A (cf. [57, Theorem 7.2]). Thus G is A-nontrivial, and all of this shows that we are in case (b). Now let us assume that G is A-trivial. Then p = 2. From Definition 4.29 we recall B(G) = {g ∈ G : ⟨g⟩ ◁ G}. From Theorem 8.11 and Definition 8.12 we know that there are two cases: (i) B(G) = G, the case of special trivial near abelian groups, and (ii) B(G) ≠ G, the case of average trivial near abelian groups. If G is a special trivial near abelian group we have exactly case (c) by Theorem 8.11. If G is an average trivial near abelian group, then it is not topologically quasihamiltonian since every compact subgroup is an average trivial near abelian group which fails to be topologically quasihamiltonian by Theorem 5.11 in [57]. For proving the converse in the nonabelian case, assume first that (b) holds. Then ⟨x, y⟩ is a standard compact near abelian subgroup in the sense of [57], for any elements x, y in G. Now, as a consequence of [57, Theorem 7.1], ⟨x, y⟩ is topologically quasihamiltonian. Therefore, by Proposition 12.3, G is topologically quasihamiltonian. When (c) holds, then G, even as an abstract group, is quasihamiltonian. Hence G is topologically quasihamiltonian. It should be noted that there may be nontrivial topologically quasihamiltonian groups for the prime p = 2 in category (b). There are, however, near abelian 2-groups which are not topologically quasihamiltonian; these are A-trivial. The following is an example. Example 12.7. [57, Ex. 4.3] Let G = ℤ(4) ⋊ ℤ(4) for the unique nontrivial automorphic action of ℤ(4) on ℤ(4). Then G is a near abelian group of order 16 which is not quasihamiltonian. The following eight-element groups are homomorphic images of G: The abelian group ℤ(2) × ℤ(4), the dihedral group D8 = ℤ(4) ⋊ ℤ(2), and the quaternion Q8 = M2 . (See [57], Example 4.3.)

12.3 The periodic case

| 233

One might notice in passing that the group A2 × M2 = A2 × Q8 occurring in Theorem 12.5 (c) is in fact hamiltonian; and it is indeed the only hamiltonian one among the whole list.

12.3 The periodic case The main goal of this section is presenting Theorem 12.11, a complete classification of the periodic locally compact topologically quasihamiltonian groups. The following result on finite quasihamiltonian groups is well known and we include the easy proof for the convenience of the reader. Proposition 12.8. A finite quasihamiltonian group is nilpotent. Proof. Suppose the statement is false. Let G be a minimum counterexample. Let p divide the order of G and x and y be elements of order p in G. Then ⟨x⟩⟨y⟩ = ⟨x, y⟩ implies that ⟨x, y⟩ has at most p2 elements, and they all have the form xa yb for a, b ∈ {0, . . . , p − 1}. If xa = yb for some a ≠ 0 and b ≠ 0, then ⟨x, y⟩ = ⟨x⟩ is cyclic, else it shows that ⟨x, y⟩ has order p2 . Thus the characteristic subset Tp of all elements of order p is a normal subgroup of G. Then, by the minimum assumption on G it follows that G/Tp is itself nilpotent. This implies that for any p-Sylow group Sp of G the quotient Sp /Tp is normal in G/Tp and hence Sp is normal in G. Thus all Sylow subgroups of G are normal and therefore G is nilpotent, a contradiction to G being a counterexample. Lemma 12.9. Let (Gj )j∈J be a family of locally compact periodic topologically quasihamiltonian groups and (Cj )j∈J , Cj ⊆ Gj , a family of compact open subgroups. Assume that the family of primes (π(Gj ))j∈J is disjoint. Then the local product G := ∏loc j∈J (Gj , Cj ) is (a) near abelian and (b) topologically quasihamiltonian.

Proof. (a) In order to show that G is near abelian, we first note that G is solvable, since the full product ∏j∈J Gj of the metabelian groups Gj is metabelian. Then G is compactly ruled by Remark 1.14. Thus Proposition 9.11 applies to show that G is near abelian if every compact open subgroup U is near abelian. Let ℱ denote the set of finite subsets of J. For each F ∈ ℱ and each j ∈ J define Gj , Kj = { Cj ,

if j ∈ F,

otherwise.

Then KF := ∏j∈J Kj is a product of locally compact near abelian groups at most a finite number of which fails to be compact. Also, KF is an open subgroup of G. We have U ⊆ G = ⋃F∈ℱ KF . Since the family of the open subgroups KF is directed and U is

234 | 12 Classifying topologically quasihamiltonian groups compact, there is one F ∈ ℱ such that U ⊆ KF . Since U and ∏j∈J Cj are open subgroups of G, there is an F 󸀠 ∈ ℱ such that F ⊆ F 󸀠 and ∏j∈J\F 󸀠 Cj ⊆ U (where we have identified in the obvious way a partial product of the Cj with a subgroup of G). Now the projection Uj of U into each Kj is compact open in Kj ; moreover, j ∈ J \ F 󸀠 implies Uj = Cj . Hence U∗ := ∏j∈J Uj is a compact open subgroup of G containing U. If we show that U∗ is near abelian, then U is near abelian by Proposition 9.4 because G is periodic. Therefore, for the remainder of the proof, we shall assume that Gj = Cj for all j ∈ J. Now we recall that for each j the group Gj is near abelian and therefore is of the form Aj Hj with a closed abelian subgroup Aj of Gj as a base subgroup, and with an inductively monothetic scaling subgroup Hj which is a standard inductively monothetic group in the sense of Definition 5.9, since G is periodic. However, since all Gj are now compact, each Hj is procyclic. Now we use the hypothesis that the {π(Gj ) : j ∈ J} is a disjoint family and conclude that H := ∏j∈J Hj is procyclic. Then A := ∏j∈J Aj is a base subgroup, H a scaling subgroup for G, and G near abelian, as we have to show. (b) For a proof of the claim that G is topologically quasihamiltonian, let X and Y be procyclic subgroups of G. We must show that XY = YX. Since the π(Gj ) form a disjoint family, X = ∏j∈J Xj with Xj = Xπ(Gj ) and, similarly, Y = ∏j∈J Yj . Therefore XY = ∏j∈J Xj Yj YX = ∏j∈J Yj Xj . Since all Gj are topologically quasihamiltonian, we have Xj Yj = Yj Xj for all j. It follows that XY = YX. Thus G is topologically quasihamiltonian by Proposition 12.3. In the following recall from Definition 2.45 that ν(G) is the set of all primes p such that every p-element commutes with every q-element for all primes q ≠ p in π(G). Lemma 12.10. All Sylow subgroups of a periodic locally compact topologically quasihamiltonian group G are normal. Equivalently, G = Gν(G) . Proof. By Theorem 2.44, we must show that every p-element x commutes with every q-element y for p and q different primes. Thus, we may replace G by L := ⟨x, y⟩. The latter group being compact and periodic, hence profinite, can be presented as the strict projective limit L = lim L/N with N running through the filter base 𝒩 of open ←󳨀󳨀N normal subgroups of L. Then, for each N, the quotient L/N is a finite quasihamiltonian group and thus [xN/N, yN/N] = 1 holds according to Proposition 12.8. Therefore [x, y] ∈ ⋂ 𝒩 = {1}. The next theorem provides a complete classification of the nonabelian periodic topologically quasihamiltonian groups and has a quick proof since it is supported by our earlier work. Recall that in the prime graph 𝒢 the primes p ∈ ν determine exactly all those lower vertices which are part of isolated vertical edges (i. e., those vertical edges which together with their endpoints are connected components of the graph).

12.4 The nonperiodic case

| 235

Theorem 12.11. Let G be a locally compact periodic topologically quasihamiltonian group. Then, for each p ∈ π(G), the set of p-elements Gp is a topologically quasihamiltonian p-group, and there is a compact open subgroup Up in Gp such that G = Gν(G) is (up to isomorphism) the local product of topologically quasihamiltonian p-groups loc

G ≅ ∏ (Gp , Up ). p∈π(G)

Conversely, every group isomorphic to such a local product is a topologically quasihamiltonian group. Proof. Lemma 12.10 implies that G = Gν(G) . Then Theorem 2.53 proves everything except the structure of the p-group factors Gp . Now each such factor Gp is a quotient group of G and therefore is a topologically quasihamiltonian locally compact p-group. Then Theorem 12.5 implies that Gp is a topologically quasihamiltonian p-group. The last statement follows immediately from Lemma 12.9. If G is a periodic locally compact topologically quasihamiltonian group, then G is near abelian hence solvable by Lemma 12.9. Theorem 12.11 shows that for each prime p the p-Sylow subgroup Sp is the direct factor Gp when G is identified with the local product, and each factor of it is identified in the obvious fashion with a subgroup. The structure of Sp was described in Theorem 12.5. For the following result we use Theorem 10.18 and Definition 10.19 when we assume that G is saturated. In particular this means that all abelian p-Sylow subgroups Sp , being central, are contained in A. Now, if G is nonabelian, then there exists at least one prime p such that Sp is nonabelian and so Sp /Ap is procyclic. By Theorem 7.15 (ii) we know (G/A)p ≅ Sp /Ap , whence G/A is Π-procyclic. So we can apply Theorem 7.36 and conclude the following. Theorem 12.12. A saturated nonabelian locally compact periodic topologically quasihamiltonian group G has a base group A for which there is a Π-procyclic scaling group H such that G = AH. We note that the full power of Theorem 7.36 is not required here since Theorem 12.11 allows the construction of H directly. From Corollary 7.38 we derive the following. Corollary 12.13. In a nonabelian periodic locally compact topologically quasihamiltonian group G, for each p ∈ π(G) the p-Sylow group is given by Sp = Ap Hp .

12.4 The nonperiodic case In Corollary 7.12 we showed that a locally compact near abelian group G which is A-nontrivial for a base group A is nonperiodic if and only if A is open and G/A is a

236 | 12 Classifying topologically quasihamiltonian groups rank 1 torsion-free group. In Theorem 7.44 we discussed an algorithmic description of such groups if a scaling subgroup was not available. We shall return to such a description below. We also studied nonperiodic locally compact near abelian groups in the context of the existence of scaling groups, for instance, in Corollary 7.26, where we showed the existence of a scaling subgroup if A is compact open. Such matters return now when we discuss them in the context of topologically quasihamiltonian groups. The first important point is that nonperiodic topologically quasihamiltonian groups are still near abelian. Let us first note an elementary fact. Lemma 12.14. Assume that S is a proper subgroup of a group G. Then ⟨G \ S⟩ = G. Proof. Let s ∈ S. Since S is a proper subgroup there is a g ∈ G \ S. Then g −1 s ∈ G \ S. Hence s = g⋅g −1 s ∈ ⟨G \ S⟩. Therefore G = S ∪ (G \ S) ⊆ ⟨G \ S⟩. Proposition 12.15. Let G be a locally compact nonabelian topologically quasihamiltonian group satisfying G ≠ comp(G). Then the following conclusions hold: (1) comp(G) is an open characteristic subgroup of G and G/ comp(G) is torsion-free of rank 1, (2) every compact subgroup of G is normal in G, (3) comp(G) is abelian, (4) G is near abelian with base A := comp(G). Proof. (1) As G is topologically quasihamiltonian, its compact elements form a subgroup of G which evidently is characteristic. Proposition 12.1 implies that G0 = {1}. Therefore there is an open compact subgroup of G showing that comp(G) is an open (and hence also closed) subgroup of G. The torsion-free discrete and quasihamiltonian group G/ comp(G) by Iwasawa’s theorem of 1943 (see [97, Theorem 2.4.11]) is of rank 1. (2) Fix any compact subgroup X of G and any element x ∈ ̸ comp(G). Then by Weil’s lemma on monothetic locally compact groups, Y := ⟨x⟩ is discrete and isomorphic to ℤ. Now XY is a closed subset of G and a subgroup since G is topologically quasihamiltonian. Let C = comp(XY); then by the preceding, C is an open normal subgroup of XY not meeting Y. Since X ⊆ C we have C = X(C ∩ Y), and since C ∩ Y = {1} we have X = C, that is, X is normal in XY. Therefore the normalizer NG (X) of X in G contains G \ comp(G). Now Lemma 12.14 allows us to conclude that every compact subgroup of G is normal in G. (3) Let 𝒩 denote the set of compact open normal subgroups of G. Since G is totally disconnected by Proposition 12.1 and every compact subgroup is normal by (2), 𝒩 is a filter basis converging to 1, and so comp(G) = lim comp(G)/N is a strict ←󳨀󳨀N∈𝒩 projective limit of discrete torsion groups comp(G)/N which are topologically quasihamiltonian by Proposition 12.1. To G/N we apply Iwasawa’s theory again (see [97,

12.4 The nonperiodic case

| 237

Lemma 2.4.8]) and conclude that each of the factor groups comp(G)/N is abelian. This results in comp(G) itself being commutative. In view of Definition 6.1, part A of near abelian groups (4) follows from (1), (2), and (3). An example with is “smallest” in some sense is the following: Example 12.16. Let s = 1+p be a unit in 1+ℤp ⊆ ℤ×p , p > 2, and consider G = ℤp ⋊ℤ with the multiplication (a, m)(b, n) = (a + sm ⋅b, m + n). Then G is a near abelian nonperiodic topologically quasihamiltonian group. Proof. In order to see that G is topologically quasihamiltonian, we may invoke Proposition 12.1 and deduce from this that it suffices to establish that G/pk ℤp ×{0} is topologically quasihamiltonian for arbitrary k ≥ 1. Now make use of [97, Theorem 2.4.11]. Example 11.13 showed that there exist abelian discrete groups in the class of nonperiodic near abelian groups which do not allow scaling subgroups. This is where our algorithmic description in Theorem 7.44 on nonperiodic near abelian groups comes into the picture. We now see how this description of nonperiodic locally compact near abelian groups applies in the topologically quasihamiltonian situation. Theorem 12.17. Let G be a nonabelian locally compact topologically quasihamiltonian group containing a discrete subgroup isomorphic to ℤ. Then the structure of G is described in Theorem 7.44 and (mod p), if 2 < p, rmp ≡ 1 { (mod 4), if p = 2.

(#)

Conversely any locally compact group satisfying with structure described in Theorem 7.44 and relations as in (#) is topologically quasihamiltonian. Proof. Let G be a topologically quasihamiltonian group; then Proposition 12.15 shows that G is near abelian and that A := comp(G) is an open abelian characteristic subgroup; at the same time it is a base. Now A = lim A/K, where K runs through a filtered ←󳨀󳨀K system of compact open subgroups of A. Since G is near abelian every K is normal in G. Therefore G = lim G/K and so it suffices to restrict our attention to discrete groups G. ←󳨀󳨀K Then A, being periodic, is a torsion group. Let us verify the congruence. Suppose that there is a prime p and a p-element a such that abm = arm,p k

and rm,p does not satisfy the congruence (#). Then there is a p-power ap of a such that k

k

[a, bm ] ∈ ̸ ⟨ap ⟩. Since the quotient group modulo ⟨ap ⟩ is finite, there is a power blm

238 | 12 Classifying topologically quasihamiltonian groups k

with [a, blm ] = 1. The quotient group of ⟨a, bm ⟩/⟨blm , ap ⟩ is a finite nonabelian quasihamiltonian p-group. Let the bar denote the passage to the quotient. Then Theorem 12.5 s ̄ implies that ā rm,p = ā bm = ā (1+p )t for some integer t. Since ā has p-power order, one deduces rm,p ≡ 1 (mod p) if p is odd and rm2 ≡ 1 (mod 4) if p = 2. Thus, we arrive at a contradiction. Conversely, assume that G is near abelian and the congruences hold. Then, in light of Proposition 12.3 it will suffice to prove that XY = YX holds for all pairs of monothetic subgroups X and Y. Knowing that G is near abelian and its subset comp(G) agrees with the base A and recalling Proposition 12.1, it suffices to assume that G is discrete. Under these circumstances it remains to prove that X and Y are permutable subgroups of G, i. e., XY = YX. This is feasible, since every subgroup and then every quotient of that subgroup of G is near abelian and monothetic subgroups X and Y satisfy the desired relations. Thus G = ⟨X ∪ Y⟩ for cyclic subgroups X and Y and G = ⟨a⟩ ⋊ ⟨b⟩ for a torsion element a and ⟨b⟩ ≅ ℤ. Observing that A := ⟨a⟩ is finite, one can obtain a natural number t with [bt , A] = 1, i. e., bt ∈ Z(G). Then X t = ⟨x t | x ∈ X⟩ and Y t are central subgroups of G of finite index. The quotient group G/X t Y t is finite and the relations then imply that it is nilpotent and a product of quasihamiltonian p-groups, i. e., G/X t Y t is quasihamiltonian. Therefore the chain of equalities XY = (XX t )(YY t ) = (XY)(X t Y t )

= (YX)(X t Y t ) = (YY t )(XX t ) = YX

implies that X and Y are indeed permutable. Hence G is topologically quasihamiltonian. Remark 12.18. If G is as in Theorem 12.17 and A = comp(G) is compact, then one knows from Corollary 7.26 that there is a discrete scaling subgroup H which is isomorphic to a nonsingleton subgroup of ℚ such that G = A ⋊ H. In any group G described in Theorem 12.15 some necessary conditions hold for G described in the following: For the sake of brevity let us write A := comp(G) in the present case. According to Remark 6.2 (ii) for an near abelian group G there is a natural homomorphism ψ : G → SAut(A) implemented by the inner automorphisms, and SAut(A) is a homomorphic image of ℤ̃ × according to Theorem 4.25, where SAut(A) = ∏p SAut(Ap ) and where SAut(Ap ) may be identified with the group of units of the ring ℛ(Ap ) of scalars of Ap , which is described in detail in Theorem 4.25. We write ψ(g) = (ψq (g))q , ψq (g) ∈ ℛ(Aq )× , q ∈ π accordingly. In Theorem 4.25, the q-primary factor (ℛ(Aq )× )q of ℛ(Aq )× is precisely described. We recall that for q = 2, the entire group ℛ(A2 )× is a 2-group.

12.4 The nonperiodic case

| 239

Lemma 12.19. Let G be a noncommutative nonperiodic nontrivial topologically quasihamiltonian group. Then for all primes q we have (q∗ )

ψq (G) ⊆ (ℛ(Aq )× )q .

Proof. We assume that (q∗ ) fails for some q and derive a contradiction. Since (2∗ ) holds trivially, we may assume q > 2. We shall produce a quotient group of G which fails to be topologically quasihamiltonian, and this will produce the required contradiction by Proposition 12.8. Firstly, since A = Aq × Bq with a complementary factor Bq by Braconnier’s Theorem 3.3, we can factor Bq and assume now that A = Aq . Secondly, let us abbreviate M = ℛ(A)× . Then we have a primary decomposition M = Mq × ∏p|(q−1) Mp by Theorem 4.25. By our assumption, there is some g ∉ A such that ψ(g) ∈ Mp \ {1} for some p|(q − 1) and so there is an a ∈ A such that gag −1 = ψ(g)(a) = s⋅a for a nonidentity scalar s ∈ Mp . So the subgroup ⟨a⟩⟨g⟩ is a semidirect product satisfying the conditions of G in view of Proposition 12.1. Thus we may now assume that A is a monothetic q-group and ⟨g⟩ ≅ ℤ, that is, we may assume that G is either G ≅ ℤq ⋊ ℤ or G ≅ ℤ(qm ) ⋊ ℤ for some m ∈ ℕ. Thirdly, since A is not divisible, we have q⋅A ≠ A. Then N = q⋅A is compact normal in G, and if G is topologically quasihamiltonian, so is G/N by Proposition 12.1 again. So we now assume A = ℤ(q), we consider ℤ(q) as the prime field of q elements, and ψ(g) ∈ Mp . So ⟨g⟩ contains a cyclic subgroup mapping onto the socle of Mp . We may replace ⟨g⟩ by this subgroup and assume that Mp has p elements and ψ(⟨g⟩) = Mp . Fourthly, we finally factor ⟨g⟩ ∩ ker ψ and now obtain a finite quotient to be identified with H := ℤ(q) ⋊ ℤ(p) where now ψ(ℤ(p)) is the cyclic subgroup of order p in ℤ(q)× . But now, on the one hand, H is still quasihamiltonian by our reduction process via Proposition 12.1, while on the other hand H is a nonnilpotent finite quasihamiltonian group, contradicting Proposition 12.8. This is a point to pause and reflect what was discussed about locally compact topologically quasihamiltonian groups in this section up to this point. Let G be a nonablian nonperiodic topologically quasihamiltonian group and let A denote the open periodic base group comp(G) according to Proposition 12.15. Assume the following hypothesis (scal) There is a scaling group for G, that is, G = AH. By Proposition 12.15(1) this means that we may assume G = A⋊H where H is a subgroup of the discrete group ℚ. If A happens to be compact, (scal) holds according to Theorem 1.33. The morphism ψ: G = A ⋊ H → SAut(A) is given by ψ(a, h) = ϕ(h) for the unique “restriction” ϕ: H → ℛ(A) = ∏ ℛ(Ap ), p

ϕ(h) = (ϕp (h))p∈π ,

240 | 12 Classifying topologically quasihamiltonian groups and the multiplication of G may be written (∀a1 , a2 ∈ A, h1 , h2 ∈ H) (a1 , h1 )(a2 , h2 ) = (a1 + ϕ(h1 )⋅a2 , h1 + h2 ). By Remark 12.18, we have (q∗∗ )

ϕp (H) ⊆ ℛ(Ap )p .

Now let K = ∏p Kp ⊆ ∏p ℤp = ℤ̃ be the profinite completion (see e. g. [93, Section 3.2]) of the discrete group H ⊆ ℚ and let κ: H → K denote the compactification morphism. Since ℛ(A) ⊆ ℤ̃ is profinite, the universal property of K gives us a unique extension ψ: G = A ⋊ K → ℛ(A),

ψ(a, k) = ϕ(k),

and (q∗∗∗ )

ϕp (K) ⊆ ℛ(Ap )p .

Now we form G∗ = A ⋊ K with multiplication (∀a1 , a2 ∈ A, k1 , k2 ∈ K) (a1 , k1 )(a2 , k2 ) = (a1 + ϕ(k1 )⋅a2 , k1 + k2 ), and the function α: A ⋊ H → A ⋊ K, α(a, h) = (a, κ(h)). The group G∗ satisfies the hypotheses of Theorems 12.5(b) and 12.11 and thus is a periodic locally compact group with open base subgroup A ≅ comp(G). We summarize the comparison of G = A ⋊ H and G∗ = A ⋊ K: Proposition 12.20. Let G denote a nonabelian nonperiodic locally compact topologically quasihamiltonian group with base group comp G and a scaling group H. Then there is an injective morphism with dense image α: G → G∗ into a locally compact periodic topologically quasihamiltonian group G∗ according to Theorem 12.5(b) and Theorem 12.11 with base group comp(G) and a scaling group isomorphic to the profinite completion of H. Specifically, α fixes comp(G) elementwise. In terms of Example 12.16, the nonperiodic topologically quasihamiltonian group ℤp ⋊ℤ is densely injected into the periodic topologically quasihamiltonian group ℤp ⋊ ℤp . Our Theorem 12.17 explains what happens if no scaling subgroup might exist.

13 Locally compact groups with a modular subgroup lattice A few historical remarks are in order. It was R. Dedekind who in 1877 proved the modular law for a certain abelian group (see [24]). However, his proof works for any abelian group. Then, in the thirties of the last century, Ø. Ore suggested to classify groups by properties of the lattice of subgroups (see [86, 87] and the historic remarks in [88]). In this spirit Iwasawa in the 1940s found his classification of quasihamiltonian groups; see [61, 62]. R. Schmidt completed this classification in [96]. For topological groups it was mainly Mukhin who tried to get analogs of Ore’s and Iwasawa’s results. In this section we present results entirely due to Mukhin (see particularly [78])— and provide new, somewhat simpler proofs by invoking our theory of near abelian groups. The subgroup lattice L(G) of a topological group is its set of closed subgroups endowed with join given as A ∨ B := ⟨A ∪ B⟩ and meet as A ∧ B := A ∩ B for A and B any closed subgroups. Then G is a topologically modular group provided the modular law holds for any closed subgroups A, B, and C of G with A a subgroup of C, i. e., A ∨ (B ∧ C) = (A ∨ B) ∧ C. Note that a group is modular if and only if its lattice of closed subgroups does not contain a sublattice isomorphic to E5 (cf. [97, Theorem 2.1.2]). ∙ ∙ ∙ ∙ ∙

13.1 Generalities Remark 13.1. The absence of E5 in the subgroup lattice is inherited by closed subgroups and quotient groups. Hence closed subgroups and quotient groups of a topologically modular group are topologically modular groups. Moreover, a locally compact ̂ is topoloabelian group G is topologically modular if and only if its Pontryagin dual G gically modular (the latter fact follows from applying the annihilator mechanism; see [54, p. 314]). However, the class of topologically modular groups fails to be closed under the formation of strict projective limits and local products as the following example, due to Mukhin, shows (see [78]). https://doi.org/10.1515/9783110599190-013

242 | 13 Locally compact groups with a modular subgroup lattice Example 13.2. Let p be any prime and I any infinite set (e. g., I = ℕ), set E := ℤ(p), define Gj := E 2 , Cj := {0} × E for all j ∈ I, and set loc

G := E (I) × E I ≅ ∏(Gj , Cj ), j∈I

where we took the discrete topology on the direct sum E (I) and the product topology I I 2 I on E I . We shall identify G with ∏loc j∈I (Gi , Ci ) and E × E with (E ) . The natural injection 2 I I I ι: G → (E ) = E × E is continuous but is not an embedding, since it is not open onto its image. Let D := {(x, x) : x ∈ E} ⊆ E 2 and I

Δ = DI = {(xj , xj )j∈I : xj ∈ E} ⊆ (E 2 ) ≅ E I × E I denote the respective diagonals. Then Δ is a closed subgroup of (E 2 )I and so ι−1 (Δ) = D(I) is a closed subgroup of G. We shall denote it by Y. This is a noteworthy and perhaps slightly unexpected fact in view of the density of E (I) in E I . We verify as an exercise that the subgroup Y is not only closed, but even discrete, since ι(Y) meets trivially every open subgroup {0} × E K for a cofinite subset K ⊆ I. Now the product E I is the strict projective limit of its finite partial products E F as F ranges through the directed set ℱ of finite subsets F of I. Accordingly, G ≅ E (I) × lim E F ≅ lim (E (I) × E F ). F∈ℱ

F∈ℱ

I

Let D2 = {(xj )j∈I ∈ E : (∃c ∈ E)(∀j ∈ I) xj = c}. Now we consider the following subgroups of G: X := E (I) × {0}, Z := E (I) × D2 , whence X ⊆ Z. Then X ∨ Y = E (I) × E I = G and so (X ∨ Y) ∩ Z = Z on the one side, while Y ∧ Z = Y and so X ∨ (Y ∧ Z) = X ∨ Y = G. Hence X ∨ (Y ∧ Z) ≠ (X ∨ Y) ∧ Z. Therefore G is a locally compact abelian nonmodular group. The example also shows that a local product of a collection of finite abelian M-groups may fail likewise to be topologically modular. Moreover, G is topologically quasihamiltonian, because it is abelian; but it is not topologically modular. Moreover, the class of topologically modular groups fails to be closed under the formation of strict projective limits and (local) products as the following example, due to Mukhin, shows (see [78], which will be reproduced in Example 13.6). Remark 13.3. Every locally compact abelian group G that is not topologically modular must contain subgroups A ⊆ C and B with B ∧ C a proper subgroup of A and C a proper subgroup of A ∨ B. Then the five closed subgroups A ∨ B, C, A, B, B ∧ C

(†)

13.1 Generalities | 243

form a subgroup sublattice of the lattice of closed subgroups of G or, equivalently, the five groups in equation (†) are all pairwise different and B ∩ C ⊆ A. Indeed, if the closed subgroups X ⊂ Z and Y do not satisfy the modular identity, then A := X ∨ (Y ∧ Z), B := Y, and C := (X ∨ Y) ∧ Z serve the purpose. This observation provides a simple method for exhibiting important examples of locally compact abelian groups not topologically modular. Example 13.4. Let G := ℝ be the reals and fix subgroups C := ℤ, A := 2ℤ, and B := √2ℤ = {z √2 : z ∈ ℤ}. Then A ∨ B = ℝ by the density of 2ℤ + √2ℤ = 2(ℤ + √2 ℤ). 2 Moreover, B ∧ C = {0} is contained in A. Example 13.5. Let p be a prime and G = Z⊕K be the topological direct sum of a discrete group Z ≅ ℤ and an infinite compact monothetic group K. Suppose that K = pK, i. e., K is p-divisible. Fix a topological generator k of K and a generator t of Z. Let C := Z, A := pZ, and B := {zt + zk : z ∈ ℤ} and observe that it is the graph of the homomorphism f : Z → K sending zt → zt +zk. Hence B is discrete and certainly B∩C = {0}. For proving G = A∨B observe first that pK = K implies that ⟨pk⟩ = K. Then A + B contains all elements of the form put + v(t + k) = put + vt + vk for u and v in ℤ. Select v := −p in order to see that pk ∈ A + B. Therefore A ∨ B = A + B contains ⟨pk⟩ = K and consequently, for every u and v in ℤ, the element put + vt belongs to A + B. Selecting u := 0 and v := 1 shows t ∈ A + B and hence ⟨t⟩ = C ⊆ A + B. Therefore A ∨ B ⊇ K ∨ C = G. Example 13.6. Let S := ℤ(p)(ℕ) and P := ℤ(p)ℕ and form G := S ⊕ P, the topological direct sum. Let ι : S → P be the canonical dense embedding of S in P and K ≅ ℤ(p) a finite subgroup of P intersecting ι(S) trivially. Such K can be provided by the subgroup of all on ℕ constant maps to ℤ(p). Define closed subgroups C := S ⊕ K, A := S, and B := {(s + ι(s)) : s ∈ S}. Then B is algebraically and topologically isomorphic to the graph of the function ι and hence a discrete subgroup of G. Then, for x to belong to B ∩ C it is necessary and sufficient that there are s, s󸀠 ∈ S, and k ∈ K with x = s + ι(s) = s󸀠 + k. Since K ∩ ι(S) = {0} we must have ι(s) = s = k = 0. Hence B ∧ C = {0}. Since A + B = S + (S + ι(S)) = S + ι(S) and ι(S) = P, one finds A ∨ B = A + B = S + P = G. Example 13.7. Let us show that the local product L := (ℤ(p2 ), pℤ(p2 )) cannot be topologically modular.

loc, ℕ

244 | 13 Locally compact groups with a modular subgroup lattice Suppose by way of contradiction that it is topologically modular. Then, for I any infinite subset of the index set ℕ with infinite complement J := ℕ \ I, we have a topological and algebraic isomorphism L ≅ LI ⊕ LJ , where LI := (ℤ(p2 ), pℤ(p2 ))loc, I and LJ := (ℤ(p2 ), pℤ(p2 ))loc, J are both algebraically and topologically isomorphic to L. The socles SI and SJ of respectively LI and LJ are compact and open therein and isomorphic to ℤ(p)ℕ . Since LI /SI ≅ ℤ(p)(ℕ) the subquotient LI /SI ⊕SJ of L is algebraically and topologically isomorphic to ℤ(p)(ℕ) ⊕ℤ(p)ℕ , which is not topologically modular, as has been shown in Example 13.6. Lemma 13.8. Let C be a compact monothetic group not a torsion group. Then there is a prime p and a monothetic subgroup K of C with pK = K and K is not a torsion group. Proof. If the connected component C0 of C is not trivial we may choose K := C0 . Since C0 is divisible, for any prime p, pK = K. Since the weight of C0 does not exceed the weight of C infer from [47, (25.17) Theorem] that C0 is monothetic. Next assume C0 = {0}. Then C is profinite and hence C = ∏p Cp is the cartesian product of its p-Sylow subgroups. If there is a prime p with Cp = {0} then K := C serves the purpose. Now assume that Cp ≠ {0} holds for all primes p. Select any prime p and note that by [54, Corollary 8.9] the closed subgroup K := ∏q=p̸ Cq cannot be torsion. Certainly pK = K. Lemma 13.9. Let G be a locally compact abelian topologically modular group. Then (a) The connected component G0 of G is compact. (b) If G contains a discrete subgroup Z ≅ ℤ, then tor(G) = comp(G) is open in G. (c) If U is any open compact subgroup, then G/U has finite ℤ-rank. Proof. (a) Since G is topologically modular, so is, by Remark 13.1, the connected component G0 . By the vector splitting theorem (see [54, Theorem 7.57]) there is n ≥ 0 and a compact connected subgroup K such that G0 = ℝn ⊕ K. If G0 were not compact then n > 0 and hence there is a closed subgroup R ≅ ℝ of G0 which must be topologically modular, contradicting the findings in Example 13.4. Hence G0 = K is compact. (b) Suppose G to contain a compact element c not torsion. Then C := ⟨c⟩ is an infinite monothetic subgroup. Lemma 13.8 provides a prime p and a monothetic infinite subgroup K of C with K = pK. Remark 13.1 shows that the closed subgroup Z ⊕ K must be topologically modular. This leads to a contradiction in light of Example 13.5. (c) If, for some open compact subgroup U of G, the factor group G/U has infinite ℤ-rank, then G contains a closed subgroup S ≅ ℤ(ℕ) ⊕ U. By (b) U is a compact torsion

13.2 The p-group case

| 245

group and thus has finite exponent (see [54, Corollary 8.9]). Since U is assumed to be infinite (else G would be discrete) there is p ∈ π(U) with Up /pUp infinite. Since G is by assumption topologically modular, so is R := S ⊕ Up /pUp . Let (Vi )i∈ℕ be a properly descending sequence of open subgroups of Up /pUp and V := ⋂i≥1 Vi denote the intersection. Then (Up /pUp )/V is first countable and has exponent p. Therefore (Up /pUp )/V ≅ ℤ(p)ℕ and hence topologically and algebraically R/V ≅ ℤ(p)(ℕ) ⊕ ℤ(p)ℕ . On the other hand, R/V, being topologically modular, contradicts our findings in Example 13.6. The Tarski monster of Remark 1.5 is a simple finitely generated group in which every proper subgroup is isomorphic to ℤ(p). Its subgroup lattice therefore cannot contain E5 and thus is modular. In view of this remark, for a classification of locally compact topologically modular groups, we shall retain our hypothesis that G be compactly ruled. In a discrete environment, for the discussion of discrete M-groups G, a similar consideration has led K. Iwasawa to the assumption that G be locally finite (see, e. g., [97]).

13.2 The p-group case It has been proved (see [57, Theorem 7.1]) that every compact topologically quasihamiltonian p-group is near abelian. Mukhin, in [78], has characterized the abelian topologically modular groups. Here we present what can be said generally about p-groups. We recall from Theorem 12.5 that a topologically quasihamiltonian p-group is always near abelian. 13.2.1 Abelian p-groups Let us, for the sake of completeness, describe the structure of an abelian topologically modular p-group. Recall from Definition 3.89 the notion of p-rank. We essentially present Mukhin’s proof and include material from [46]. A fact about certain p-groups of exponent p2 and the local product L := (ℤ(p2 ), pℤ(p2 ))

loc, ℕ

(*)

will be needed. Lemma 13.10. Assume that a locally compact abelian p-group G has a descending basis of 0-neighborhoods of compact open subgroups V1 ⊇ V2 ⊇ V3 ⊇ ⋅ ⋅ ⋅ and a family {Zk : k ∈ ℕ} of subgroups with isomorphisms ζk : ℤ(p2 ) → Zk satisfying the following conditions:

246 | 13 Locally compact groups with a modular subgroup lattice (a) V1 has exponent p, (b) The sum Z1 +⋅ ⋅ ⋅+Zm +Vm+1 is direct (algebraically and topologically) for m = 1, 2, . . . , (c) p⋅Zk ⊆ Vk for k = 1, 2, . . . . Set FG := ∑k∈ℕ Zk and LG := FG ; further PG := p⋅FG ⊆ V1 and observe LG = FG + PG . Then there are: (i) an algebraic isomorphism ηF : ℤ(p2 )(ℕ) → FG such that the restriction to the k-th summand is ζk ; (ii) an isomorphism of compact groups ηP : pℤ(p2 )ℕ → PG such that the restriction to the kth factor is ζk |pℤ(p2 ); and (iii) an isomorphism of topological groups ηL : L → LG . Proof. There is no loss of generality to assume G = LG . For each k we have an isomorphism ζk : ℤ(p2 ) → Zk by the definition of Zk . Conclusion (i) then follows from Assumption (b) . Since V1 has exponent p and FG ∩ V1 is dense in V1 , from (c) we conclude FG ∩ V1 = pFG , i. e., PG = V1 . So PG is open in G. As a consequence by the density of FG in LG we have G = FG + PG . Furthermore, (∀k ∈ ℕ) Zk ∩ PG = Zk ∩ V1 ∩ FG = Zk ∩ pFG = pZk . We now prove (ii). Let us set FG,k := ⨁1≤j≤k Zj . Then (b) and (c) imply PG = pFG,k ⊕ Vk+1 , that is, there is a projection pk : PG → pFG,k , and for each k ≥ 1 there is a canonical projection ϕk : pFG,k → pFG,k−1 with kernel pZk . So (pFG,k , ϕk )k∈ℕ forms an inverse system with projective limit lim pZG,k = ∏k≥1 pZk . By the universal property of the ←󳨀󳨀k limit, there is a unique morphism ϕ: PG → lim pFG,k = ∏ pZk . ←󳨀󳨀 k k

Since all morphisms PG → pZm are surjective, so is ϕ and since these morphisms separate the points, ϕ is an isomorphism of compact groups. By the definition of Zk we have an isomorphism α: pℤ(p2 )ℕ → ∏k≥1 pZk so that the restriction and corestriction to the kth factor agrees with ζk |pℤ(p2 ) : pℤ(p2 ) → pZk . Thus ηP = ϕ−1 ∘ α: pℤ(p2 )ℕ → PG is an isomorphism as claimed mapping the kth factor of pℤ(p2 )ℕ to pZk ⊆ PG . For a proof of (iii) we set F := ℤ(p2 )(ℕ) and let ηF : F → FG be the unique homomorphism that agrees with ζk on the k-th component. Take an element (zn )n∈ℕ ∈

13.2 The p-group case

| 247

ℤ(p2 )(ℕ) ∩ pℤ(p2 ). Then ηF (z) = ∑n∈ℕ ζn (zn ) by (i) in view of the definition of a direct sum. If we identify the dense direct sum ∑n∈ℕ Zn in the obvious fashion with a subgroup of the product PG according to (ii), then we also have ηP (z) = ∑n∈ℕ ζn (zn ) by (ii). Hence ηF and ηP agree on F ∩ P and thus define a unique algebraic morphism η: L = F +P → FG +PG = G. Since η agrees on the open subgroup P with the continuous and open map ηP it is continuous and open. Since ηF is an isomorphism, FG is in the image of η. Similarly, PG is in the image of η as well. Hence η is surjective. If η(z) = 0 and z ∈ F, then 0 = η(z) = ηF (z) implies z = 0 since ηF is injective. Similarly z = 0 follows if z ∈ P. Therefore η is injective and thus is an isomorphism of topological groups. This completes the proof. Theorem 13.11 (Mukhin, see [78]). A locally compact abelian topologically modular p-group G satisfies one of the following conditions: (a) G contains an open compact subgroup of finite p-rank; then the torsion subgroup T = tor(G) of G is discrete and G/T has finite p-rank. (b) There is an open compact subgroup U of G with infinite p-rank; then G/U has finite p-rank and G contains a closed subgroup D of finite p-rank with compact factor group G/D. In particular, D can be taken to be div(G). Remark. Since at a later stage Theorem 14.9 will show that either Condition (a) or (b) implies G to be strongly topologically quasihamiltonian. It therefore is our task to prove that any topologically modular p-group satisfies (a) or (b). Proof. Suppose first that the premise of (a) is valid, i. e., rankp (U) is finite for some open compact subgroup U of G. Then, if necessary, we may take Proposition 3.92 into account and replace U by one of its open subgroups and thus assume that U is torsionfree. Therefore tor(G) must be a discrete subgroup. As the torsion-free group G/ tor(G) contains the open subgroup (U + tor(G))/ tor(G) and the latter has finite p-rank, Lemma 3.93 implies that G/T = G/ tor(G) has finite p-rank. Let us assume the premise of (b) now. Suppose, by way of contradiction, the p-rank of G/U to be infinite. We shall derive a contradiction from this by showing that a local product isomorphic to the one in equation (*) can be manufactured to be a factor group of a closed subgroup of G, being topologically modular by Remark 13.1, and then to refer to Example 13.7. Claim 1: One can assume U to have exponent p. With G also G/pU is topologically modular. Consider instead of G and U the factor group G/pU and its open compact subgroup U/pU which still has infinite p-rank. Claim 2: One can assume U to be first countable and hence metric. Moreover U ≅ ℤ(p)ℕ . There is a strictly decreasing sequence (Vk )k≥1 of open subgroups of U. Letting V := ⋂k≥1 Vk we pass from (G, U) to (G/V, U/V). Then G/V is topologically modular and U/V is infinite and therefore has infinite p-rank. Moreover U/V is first countable. The statement follows then from U having exponent p.

248 | 13 Locally compact groups with a modular subgroup lattice Claim 3: One can assume that pG ∩ U is open in G. If T := pG ∩ U is not open in G it must have infinite index in U. The factor group G/T has exponent p and, as G/U is infinite, so is (G/T)/(U/T). Since G/T may be considered a GF(p)-vector space the open subgroup U/T admits a complement, say S. Thus algebraically and topologically G/T ≅ S ⊕ U/T. Selecting in G/T a countable subgroup Σp ≅ ℤ(p)(ℕ) and observing that U/T, by claim 2, is topologically isomorphic to ℤ(p)ℕ , it turns out that Σp ⊕ U/T is a factor group of a subgroup of G and therefore topologically modular by Remark 13.1. This contradicts the finding in Example 13.6. Claim 4: There is a sequence (Fk , Vk )k≥1 of pairs of closed subgroups of G where for all k ≥ 1 (1) Fk is finite, Vk is open, and Fk ∩ Vk+1 = {0}; (2) Fk ∩ Vk = ⟨pxk ⟩ for some nontrivial element xk ∈ G; and (3) Fk ⊆ Fk+1 , Vk+1 ⊆ Vk , and ⋂k≥1 Vk = {0}. We proceed by induction on k. For k = 1 let V1 be the open subgroup pG ∩ U (see claim 3). Then there exists x1 ∈ G of order p2 with px1 ∈ V1 . Set F1 := ⟨x1 ⟩. Suppose (Fi , Vi ) have been found for 1 ≤ i ≤ k. Then Fk is finite and hence there exists an open subgroup Vk+1 contained in Vk with Fk ∩ Vk+1 = {0}. Since pG ∩ V1 is dense in V1 by claim 3 the intersection pG ∩ Vk+1 is dense in Vk+1 . Therefore one can find xk+1 ∈ G of order p2 with pxk+1 ∈ Vk+1 . Set Fk+1 := Fk ⊕ ⟨xk+1 ⟩. (1), (2), and the first statement of (3) are now clear from the construction. Selecting at each step Vi properly one can achieve ⋂i≥1 Vi = {0} as desired. Setting in claim 4 for all k ∈ ℕ respectively Zk := ⟨xk ⟩ and F := ⟨Zk : k ≥ 1⟩ shows that the assumptions of Lemma 13.10 hold. Therefore there is a closed subgroup L of G topologically and algebraically isomorphic to the group in equation (*). We have reached a contradiction and therefore the p-rank of U must be finite. For proving the remaining assertions let us return to the original meaning of G and its open compact subgroup U of infinite p-rank. By what we just proved, the factor group G/U has finite p-rank. Thus, according to Proposition 3.92, G/U ≅ ℤ(p∞ )m ⊕ F, for some m ≥ 0 and finite group F. Replacing U by the preimage of F in G allows to ̂ must contain have F = {0}. Using Pontryagin duality one concludes that the dual G m ̂ satisfies the premise of (a) and therefore an open compact subgroup ≅ ℤp . Hence G ̂ ̂ tor(G) is a closed discrete subgroup of G. Using duality again, one concludes that for ̂ ⊥ the factor group G/D is compact. D := tor(G) Since G/D is reduced, D contains the maximal divisible subgroup div(G) of G. Therefore div(D) has finite p-rank and thus agrees with div(D). As G/(U + div(D)) is a reduced topologically modular group of finite p-rank, it is compact. Hence G/ div(G) is compact.

13.3 Iwasawa (p, q)-factors |

249

13.2.2 Nonperiodic locally compact abelian groups Here we deal with nonperiodic locally compact abelian groups. Such groups either are totally disconnected but not periodic or have nontrivial connected components. Section 14.5.2 discusses groups with nontrivial connected components. Theorem 13.12. Let G be a totally disconnected nonperiodic nondiscrete locally compact abelian group. If G is topologically modular, then all of the following conditions hold: (a) tor(G) is an open subgroup and G/ tor(G) has finite ℤ-rank, (b) there is a finite subset ϕ of π(tor(G)) such that for κ := π(tor(G)) \ ϕ the κ-primary subgroup tor(G)κ is discrete; the p-component Tp of tor(G) is a discrete subgroup of G, (c) for every p ∈ π(tor(G)) the primary component Tp is topologically modular. Proof. (a) Lemma 13.9 (b) implies the openness of tor(G). Suppose that G/ tor(G) has infinite ℤ-rank. Then G must contain a discrete subgroup S ≅ ℤ(ℕ) . Since, by assumption, G is not discrete, any open compact subgroup must be infinite. Fix any such subgroup C of tor(G) and apply Lemma 13.9 (c) in order to see that (S ⊕ C)/C should have finite rank, contrasting S having infinite ℤ-rank. Hence (a) is established. (b) Let U be any open compact subgroup of tor(G). Since U has finite exponent (see [54, Corollary 8.9]), deduce that ϕ := π(U) is finite and that (tor(G))κ has trivial intersection with U and hence is discrete. (c) Since every closed subgroup of a topologically modular group is topologically modular Tp must be topologically modular. Corollary 13.13. Let the locally compact abelian group G satisfy (a)–(c) of Theorem 13.12. Then every torsion-free subgroup is discrete and hence closed. Moreover, every topologically finitely generated subgroup is discrete and hence closed. Proof. Let H be a torsion-free subgroup of G. Then its closure H evidently satisfies conditions (a)–(c). Thus tor(H) is open and, since H ∩ tor(H) = {0}, deduce that H must be a discrete subgroup of H. The second statement is proved similarly.

13.3 Iwasawa (p, q)-factors Let us give a name to locally compact topologically modular p-groups and reveal their structure as a consequence of Lemma 13.20. Definition 13.14. A locally compact topologically modular p-group shall be called an Iwasawa p-factor. The number p is called the prime of the factor.

250 | 13 Locally compact groups with a modular subgroup lattice In this subsection we describe certain candidates of topologically modular groups. In Theorem 13.31 it will turn out that they, together with the Iwasawa p-factors, are the building blocs of any compactly ruled topologically modular group. Example 13.15. For a prime q let A be an additively written locally compact abelian group of exponent q according to Corollary 3.26. Thus, algebraically, A is a vector space over the field GF(q). Now let p be a prime such that p|(q − 1). Then the multiplicative group of GF(q) contains a cyclic subgroup Z of order p. Let C = ⟨t⟩ be any p-procyclic group (that is, C ≅ ℤ(pk ) for some k ∈ ℕ or C ≅ ℤp ) and let ψ: C → Z be an epimorphism. Then C acts on A via r∗a = ψ(r)⋅a. Since Z is of order p, the kernel of ψ is an open subgroup of C of index p. Set G = A ⋊ψ C, the semidirect product for the action of C on A. Then A := A × {1} is a base subgroup of the near abelian locally compact group G, and H = ⟨(0, t)⟩ = {0}×C is a procyclic scaling p-subgroup. There are many maximal p-subgroups of G, namely, each ⟨(a, t)⟩ for any a ∈ A, and there is one unique maximal q-subgroup which is normal, namely, A. The simplest case arises when we take for C the unique cyclic group Sp (Z) of Z of order p, in which case we have G ≅ A ⋊ ℤ(p) and the set of elements of order p is A × (ℤ(p) \ {0}) and the set of q-elements is A × {0}. The class of locally compact near abelian topologically modular groups described in Example 13.15 is relevant enough in our classification to deserve a name. Definition 13.16. A locally compact group G which is isomorphic to a semidirect product A ⋊ψ C as described in Example 13.15 will be called an Iwasawa (p, q)-factor. The primes p and q are called the primes of the factor G. The prime graph 𝒢 of an Iwasawa (p, q)-factor is one sloping edge epq with its endpoints; see Definition 10.1. Let, for a group H and prime p, the subgroup topologically generated by all pth powers of elements in H be denoted by H p . We have an abstract characterization of Iwasawa (p, q)-factors as follows. Proposition 13.17. Let p and q be primes satisfying p|(q − 1). A near abelian group G is an Iwasawa (p, q)-factor if and only if it satisfies the following conditions: (a) A = G󸀠 is a locally compact abelian group of exponent q; it is either a compact or a discrete subgroup of G; (b) there is a scaling group H which is a procyclic p-group; it induces an action of order p on A. If these conditions are satisfied, then G = A ⋊ H is a semidirect product and Z(G) = H p .

13.3 Iwasawa (p, q)-factors | 251

Proof. Any group satisfying (a) and (b) is isomorphic to A ⋊ H; (b) implies that the kernel of the action of H on A is open in H of index p. Moreover, A identifies with A × {1}. Thus G is an Iwasawa (p, q)-factor according to Example 13.15. Conversely, assume that G = A ⋊ C is an Iwasawa (p, q)-factor. Then A := A × {1} is an abelian group of exponent q and H := {0} × C acts by conjugation on A. Since this action is scalar of order p ≠ q on A one deduces from standard commutator relations G󸀠 = (AH)󸀠 = A󸀠 [A, H]H 󸀠 = [A, H] = A. This yields G = AH, satisfying conditions (a) and (b). Proposition 13.18. Let G = AH be an Iwasawa (p, q)-factor and A be a topologically modular group. Then G is a topologically modular group. Proof. As noted, H is a procyclic p-group and A is like in Proposition 13.17 (a). We need to establish the modular identity X ∨ (Y ∧ Z) = (X ∨ Y) ∧ Z for any closed subgroups X, Y, Z of G where X is contained in Z. Since G is near abelian and closed subgroups are near abelian, by Lemma 7.39, there are decompositions X = AX HX ,

Y = AY HY ,

Z = AZ HZ

and one can arrange HX ≤ HZ and AX ≤ AZ . Since A, by assumption, is a topologically modular group the identity AX ∨ (AY ∧ AZ ) = (AX ∨ AY ) ∧ AZ holds, and AX ∨ (AY ∧ AZ ) is normal in G and we may factor it. Hence, in the sequel, we can assume that AX = AY ∩ AZ = {1}. Let us observe that H p = ⟨hp | h ∈ H⟩ is the unique p-subgroup of AH p and that H p is procyclic. Next, let us observe that L := ⟨H, HZ ⟩ is finitely generated and hence a profinite near abelian group. Since HZ is a p-subgroup of L it is conjugate to a subgroup of the p-Sylow subgroup H, i. e., there is x ∈ L with HZx ≤ H. Replacing thus X, Y, and Z by their respective conjugates with x, we can arrange that HZ ≤ H. Therefore the modular identity, which we need to prove, now reads (HX ∨ AY HY ) ∧ AZ HZ = HX ∨ (AY HY ∧ AZ HZ ). If HX = {1} this is indeed an identity. Hence, from now on, we may assume that HX ≠ {1}. Suppose next that HY = {1}. Then the modular identity reads HX AY ∧ AZ HZ = HX ∨ (AY ∧ AZ HZ ). By our arrangements HX ≤ HZ ≤ H and then, as G = A × H is a direct product, when considered as a topological space, it follows readily that HX AY = AY × HX , AZ HZ = AZ × HZ , and AY corresponds to the subspace AY × {1}. Then the identity clearly holds, as a consequence of the fact that it holds for involved subgroups when considering

252 | 13 Locally compact groups with a modular subgroup lattice topological subspaces of the direct product A × H. Thus, from now on, we can assume that HY ≠ {1}. If HX ∩ HY = H it follows from HX ≤ HZ ≤ H that HX = HY = HZ and the direct product argument just exercised shows the identity to hold, when the subgroups are considered subspaces of the direct product G = A × H. Hence we may assume that HX ∩ HY is a proper subgroup of H. Since HX ∩ HY is contained in HZ and is a normal subgroup of G, we may factor it and thus assume from now on that HX ∩ HY = {1}. This implies that H p = ⟨hp | h ∈ H⟩ = {1}, i. e., H can be only of order p and thus HX = HZ = H. Summarizing, we have achieved that HX = HZ = H, HY ≠ H, and HY = aHa−1 for some a ∈ A, since in G, all subgroups of order p form a single conjugacy class. The modular identity now reads (H ∨ AY aHa−1 ) ∧ AZ H = H ∨ (AY aHa−1 ∧ AZ H). Let h be a generator of H. Then a 󳨃→ aha−1 h−1 is injective, since the subgroup ⟨a⟩⋊⟨h⟩ is a finite Frobenius group with kernel ⟨a⟩ and complement ⟨h⟩. Therefore the left-hand side turns out to be equal to (⟨AY , a⟩ ∧ AZ )H. Analyzing the right-hand side, suppose that x ∈ (AY aHa−1 ∨ AZ H). Then there are aY ∈ AY , aZ ∈ AZ , and h, h󸀠 in H such that x = ay aha−1 = aZ h󸀠 . Considering this identity modulo A implies the equality h = h󸀠 . It follows from the above-mentioned injectivity of a 󳨃→ aha−1 h−1 that aha−1 h−1 belongs to ⟨AY , a⟩∩AZ and therefore the right-hand side turns out to be equal to (⟨AY , a⟩ ∧ AZ )H as well. Thus the modular identity holds in all cases and therefore G is a topologically modular group. Corollary 13.19. Any nondegenerate Iwasawa (p, q)-factor provides a topological Mgroup which is not quasihamiltonian. Proof. If G = AH is a nondegenerate Iwasawa (p, q)-factor according to Proposition 13.17, then Gp = H is a p-Sylow subgroup and Gq = A is a q-Sylow subgroup such that [Gp , Gq ] ≠ {1}. Hence {p, q} ≠ ν(G) and Theorem 12.11 shows that G cannot be topologically quasihamiltonian.

13.4 The periodic case The main objective of this section is to collect facts for giving a complete description of compactly ruled topologically modular groups, namely, Theorem 13.31. Lemma 13.20. Any compactly ruled topologically modular group G is near abelian.

13.4 The periodic case

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Proof. Theorem 9.18 implies that we may assume G to be finitely generated. Then G is, in particular, compact, i. e., the strict projective limit of finite modular groups. Any such finite group is near abelian, by a result of Iwasawa; see Theorem 2.4.4 in [97]. Knowing now that every compactly ruled topologically quasihamiltonian G is near abelian we can turn to classifying all those compactly ruled near abelian groups which are topologically modular groups. Lemma 13.20 tells that any Iwasawa p-factor is a near abelian p-group of the form G = AH for a base group A and a procyclic scaling subgroup H. Later, Proposition 14.24 will immediately imply that G is topologically quasihamiltonian and then Theorem 12.5 in conjunction with Theorem 14.9 allows to precisely describe the structure of A. Our first step in this direction is to recall Definition 2.45 and to obtain at once a decomposition of G. Theorem 13.21. Let G be a locally compact compactly ruled topologically modular group. Then, for each p ∈ π(G), the set of p-elements Gp is an Iwasawa p-factor, and there is a compact open subgroup Up in Gp such that G = Gν(G) × Gν(G)󸀠 and Gν(G) is (up to isomorphism) the local product of Iwasawa p-factors, i. e., loc

∏ (Gp , Up ).

p∈π(G)

Proof. Observe that G is compactly ruled. An application of Theorem 2.53 yields the desired description. Lemma 13.22. Let (Gj )j∈J be a family of locally compact compactly ruled topologically modular groups and (Cj )j∈J , Cj ⊆ Gj , a family of compact open subgroups. Assume that the family of primes (π(Gj ))j∈J is pairwise disjoint. Then the local product G :=

∏loc j∈J (Gj , Cj ) is a topologically modular group. In particular, G is near abelian.

Proof. As said earlier, the containment X ∨(Y ∧Z) ⊆ (X ∨Y)∧Z is always true for closed subgroups X, Y, and Z with X a subgroup of Z. For proving the reverse containment, pick g ∈ (X ∨ Y) ∧ Z. Then g = (gj ) is uniquely decomposed into factors gj ∈ Gj and one has, for every j ∈ J, gj ∈ (Xj ∨ Yj ) ∧ Zj , where Xj , Yj , and Zj stand for the respective intersections of X, Y, and Z with Gj . By assumption, Gj is a topologically modular group, and hence, for each j ∈ J, gj ∈ Xj ∨ (Yj ∧ Zj ). Since Xj ∨ (Yj ∧ Zj ) is contained in X ∨ (Y ∧ Z), conclude that g itself belongs there.

254 | 13 Locally compact groups with a modular subgroup lattice Recall from Definition 2.45 that ν(G)󸀠 is the set of primes p such that for each Sylow p-subgroup there is a p󸀠 for which there is a p󸀠 -Sylow subgroup Sp󸀠 such that [Sp , Sp󸀠 ] ≠ {1}. Now we construct a locally compact topologically modular group G satisfying π(G) = ν(G)󸀠 . Example 13.23. Let A be any locally compact abelian group A allowing an injective map χ: π(A) → π, the set of all primes. For each q ∈ π(A), let Hχ(q) be a procyclic χ(q)-group acting on Aq -nontrivially by scalar multiplication so that Aq ⋊ Hχ(q) is an Iwasawa (χ(q), q)-factor. Assume that for each q ∈ π(A) there is a compact open subgroup Cq × Kq of Aq ⋊ Hχ(q) . Set loc

G := ∏ ((Aq ⋊ Hχ(q) ), Cq × Kq ). q∈π(A)

By Lemma 13.22, G is a compactly ruled locally compact topologically modular group. The base group of G is loc

A ≅ ∏ (Aq × {1}, Cq × {1}) q∈π(A)

and we identify the two isomorphic groups. The group loc

H := ∏ ({1} × Hχ(q) , {1} × Kq ) q∈π(A)

is a Π-procyclic scaling group of G such that G = AH. The prime graph 𝒢 consists of connected components each of which consists of a sloping edge eχ(q)q and its endpoints, where q ranges through π(A). Notice that – in a planar representation of the prime graph 𝒢 it is quite possible that an edge eχ(q)q crosses one or several other edges eχ(q󸀠 )q󸀠 without their intersecting in the graph theoretical sense. For instance, since 3 divides 43 − 1 and 5 divides 31 − 1, one may consider the cartesian product of the uniquely determined Iwasawa (3, 43)-factor of order 3 × 43 and the Iwasawa (5, 31)-factor of order 5 × 31. The prime graph of this topologically modular group looks as follows (the two lines do not intersect in the graph theoretical sense): 3

5 31

43.

13.4 The periodic case

| 255

For the remainder of this section, we assume the following hypothesis for the group G. (H) G is a compactly ruled locally compact topologically modular group with a base group A, a scaling subgroup H, and a prime q ∈ π(A) such that q ∈ ν(G)󸀠 = π(G). Lemma 13.24. Let G satisfy (H). Then Gq = Aq and Gq is an exponent q normal abelian subgroup. Proof. The claim is (G/A)q = {1}. Because G = AH this is implied by Hq = {1}. So we have to prove the following claim. Claim (i). Hq = {1}. Since q ∈ ν(G)󸀠 , there is an x ∈ Sq = Aq Hq and a y ∈ Gp for a p ≠ q such that [x, y] ≠ 1. Now yA ∈ G/A is a p-element since y is a p-element. Hence yA ∈ (G/A)p = AHp /A whence y ∈ AHp . Let a ∈ Aq be arbitrary. Thus X := Hp ∪Hq ∪{a, x, y} is a compact subspace of G. Since G is compactly ruled, C := ⟨X⟩ is a compact subgroup which is a topologically modular group by Remark 13.1. We observe C ⊆ AHp Hq = AHpq . Then C = C ∩ AHpq = (C ∩ A)Hpq . Thus C ∩ A is a base group of C, Hpq is a scaling group, and C satisfies (H). For a proof of claim (i) we may therefore assume that G is profinite. Now we use the fact that the lemma is true for finite modular groups by Iwasawa’s Theorem 2.4.4 in [97]. So let N be a compact open subgroup of G. Then G is a topologically modular group by Remark 13.1. Now yN and xN are p-, respectively, q-elements and neither of them is contained in N if N is small enough. Since the lemma holds for finite groups, claim (i) is true for G/N and thus we know that Hq N/N = {N}. That is, Hq ⊆ N. Since G is profinite, the intersection of all open normal subgroups is {1}. Hence claim (i) is proved. Claim (ii). The component Aq has exponent q. We continue to consider G to be profinite. Since the lemma is true for finite groups, we know that aN has order q or agrees with N. So if a ≠ 1, then aN has order q as soon as a ∉ N. In other words, aq ∈ N, and since G is profinite, aq = 1 follows. Since a was arbitrary in Aq , indeed Aq has exponent q. Lemma 13.24 says that in the prime graph 𝒢 the lower vertex belonging to q is uncovered, that is, q ∈ γ(G). (See Definition 10.2 and Definition 10.6.) Remark 13.25. In the circumstances of Lemma 13.24, Gq = Aq and Hq = {1} are equivalent. Lemma 13.26. Let G satisfy (H). Then there is a unique p = χ(q) ∈ π(G) such that p|q − 1 and Gp = Hp = ⟨hp ⟩, and the map u 󳨃→ [hp , u] : Aq → Aq is an automorphism. Proof. We assume q ∈ π(G) as stated in (H). Claim (i). There exist p and hp ∈ H such that x 󳨃→ [hp , x] : Aq → Aq is an automorphism.

256 | 13 Locally compact groups with a modular subgroup lattice As in the proof of claim (i) in the proof of Lemma 13.24 we find an x ∈ Sq = Aq (where we now know that Hq = {1} whence x ∈ Aq !) and a y ∈ Gp for a p ≠ q such that [x, y] = 1. This time we let a ∈ Aq be arbitrary. We define, in a similar fashion as before, C := ⟨Hp ∪ {a, x, y}⟩ ⊆ AHp , yielding the compact topologically modular group C = (C ∩ A)Hq satisfying (H) as in the proof of the Lemma 13.26. Let N be an open normal subgroup of the profinite group C containing neither x nor y. Then C/N is a finite topologically modular group satisfying (H), so that we can apply what is known for finite M-groups by Iwasawa’s Theorem 2.4.4 in [97]. We conclude that (C/N)pq is an Iwasawa (p, q)-factor of G/N and so aN/N = {N}. Therefore a ∈ N for all sufficiently small N, which shows a = 1. Therefore, Ap = {1}. Since at least one p-Sylow subgroup is of the form Ap Hp by Corollary 7.38 , this means that Gp = Hp = ⟨hp ⟩. So y ∈ Hp and we conclude that [hp , x] ≠ 1. Then Theorem 7.20 applies to our present situation and shows that u 󳨃→ [hp , u] : Aq → Aq is a scalar automorphism. Since this implies p|q − 1, this completes the proof of claim (i). There remains the proof of the following claim: Claim (ii). If there is a p∗ -element y∗ such that [x∗ , y∗ ] ≠ 1 for some x∗ ∈ Aq , then p∗ = p. For a proof of this fact we define C ∗ = ⟨Hp ∪ {x ∗ , y∗ }⟩. Once more using the fact that G is compactly covered we deduce that C ∗ is a profinite topologically modular group satisfying (H). We invoke again Iwasawa’s Theorem 2.4.4 of [97] in order to conclude that for any open normal subgroup N small enough so that neither x∗ , nor y∗ , nor hp is contained in N, the p∗ element y∗ N is contained in Hp N/N = (G/N)p and thus p∗ = p as asserted. Remark 13.27. In the circumstances of Lemma 13.26, Gp = Hp and Ap = {1} are equivalent. After Lemma 13.26 we know that in the prime graph 𝒢 of G the funnel pointing to the lower vertex associated with q (see Definition 4.35) contains one and only one sloping edge epq . Yet it is still possible that there is a sloping edge epq∗ to a lower vertex associated with a prime q∗ ≠ q. This needs to be ruled out by methods we now are familiar with. Lemma 13.28. Let G satisfy (H). If there is an element a∗∗ ∈ Aq∗∗ in G such that [hp , a∗∗ ] ≠ 1, then q∗∗ = q. Proof. Let 1 ≠ a ∈ Aq and form the compact subgroup C ∗∗ = ⟨Hp ∪ {a, a∗∗ }⟩. Then C ∗∗ is a profinite topologically quasihamiltonian group; we let N be any open normal subgroup not containing any of the elements hp , a, or a∗∗ . Then C ∗∗ /N is a finite modular group to which we apply Iwasawa’s Theorem 2.4.4 in [97] and conclude that a∗∗ N is an element of the Iwasawa (p, q)-factor (C ∗∗ /N)pq and thus satisfies a∗∗ N ∈ (C ∗∗ /N)q . Therefore q∗∗ = q, as asserted.

13.5 A summary of periodic locally compact topological M-groups | 257

What we have achieved so far is this: Assume that G satisfies (H). Then there is a pair (χ(p), p) of primes satisfying χ(p)|q − 1 and a closed subgroup Gχ(q)q = Aq Hχ(q) ≅ Aq ⋊ Hχ(q) .

(1)

By Proposition 13.17, Gχ(q)q is an Iwasawa (χ(q), q)-factor. Moreover, Hp acts trivially on all of H since H is abelian, and it acts trivially on all Aq∗∗ for q∗∗ ≠ q by Lemma 13.28. So it acts trivially on all r-Sylow subgroups Sr = Ar Hr for r ≠ q. By Lemma 13.20, the group G is near abelian. By (H) it is A-nontrivial and by Lemma 13.26 the factor group G/A is Π-procyclic. Therefore Corollary 7.38 applies to show that all other r-Sylow subgroups are conjugate to Sr . Moreover, the group Aq commutes elementwise with all Aq∗∗ since A is abelian. Further it commutes with all Hp∗ for p∗ ≠ p by Lemma 13.26. Therefore Aq commutes elementwise with all Sr = Ar Hr for r ≠ χ(q), and all other r-Sylow subgroups are conjugate to Sr by Corollary 7.38. This allows us to sum up this section as follows. Theorem 13.29. Let G be a compactly ruled locally compact topologically modular group with a base group A and a scaling group H such that π(G) = ν(G)󸀠 . Then there ̇ is a bijection χ: π(A) → π(H) such that π(G) = π(A)∪π(H) and that for each q ∈ π(A)

the χ(q)q-Sylow subgroup Sχ(q)q = Aq Hχ(q) ≅ Aq ⋊ Hχ(q) is normal and is an Iwasawa (χ(q), q)-factor. Different Iwasawa (p, q)-factors intersect trivially and commute elementwise. The prime graph 𝒢 (G) of G is the disjoint union of the open closed connected components consisting of the edges {eχ(q)q } and its endpoints as q ranges through π(A). We note that the normal {χ(q), q}-Sylow subgroup Sχ(q)q is the set G{χ(q)q} of all {χ(q), q}-elements of G for q ∈ π(A).

13.5 A summary of periodic locally compact topological M-groups At this time we are prepared to give an explicit and comprehensive structure theorem of a locally compact compactly ruled nontrivial topologically modular group G. 󸀠 ̇ We have π(G) = ν(G)∪ν(G) and by Theorem 13.21 loc

G ≅ ( ∏ (Gp , Up )) × Gν(G)󸀠 p∈ν(G)

(1)

for a set of normal Iwasawa p-factors Gp which are either abelian or a p-group whose structure is completely described in Theorem 12.5. Therefore we have to describe the structure of Gν(G)󸀠 , for which we refer to Theorem 13.29. Let Aν󸀠 be the base group and Hν󸀠 a scaling group of Gν(G)󸀠 . Let σ be an abbreviation for π(Aν󸀠 ) and recall our bijection

258 | 13 Locally compact groups with a modular subgroup lattice χ: σ → π(Hν󸀠 ) from Theorem 13.29. By Theorem 3.3 we have local product representations loq

Aν󸀠 = ∏(Aq , Bq ),

(2)

q∈σ loc

Hν󸀠 = ∏(Hχ(q) , Kχ(q) ) q∈σ

(3)

for suitable compact open subgroups B of Aν󸀠 and K of Hν󸀠 . We obtain a compact open subgroup C = BK of Gν(G)󸀠 = Aν󸀠 Hν󸀠 ≅ Aν󸀠 ⋊ Hν󸀠 . Componentwise multiplication of (2) and (3) is possible after Theorem 13.29, i. e., loq

loq

q∈σ

q∈σ

Gν(G)󸀠 = ∏(Aq Hχ(q) , Bq Kχ(q) ) = ∏(Gχ(q)q , Cχ(q)q ).

(4)

It is practical to introduce a uniform set of indices for this situation. Definition 13.30. For a locally compact compactly ruled near abelian topologically modular group G with a base group A we define a set I(G) of subsets of π(G) as follows: I(G) = {{p} : p ∈ ν} ∪ {{p, q} : q ∈ π(Aν(G)󸀠 ), p = χ(q)}. Then we have the following summary for (1)–(4). Theorem 13.31 (The main theorem on topological M-groups). Let G be a compactly ruled locally compact topologically modular group. Then there is a unique family (Gσ )σ∈I(G) of normal σ-Sylow subgroups which commute elementwise such that for a family of compact open subgroups Cσ ⊆ Gσ , the group G is represented as their local product, i. e., loc

G ≅ ∏ (Gσ , Cσ ). σ∈I(G)

The groups Gσ are Iwasawa p-factors for σ = {p} ∈ I(G) and are Iwasawa (p, q)-factors for σ = {p, q} ∈ I(G). Conversely, every G of this form is a topologically modular group. Proof. While the proof for G having the given description is contained in the text preceding the theorem, we need to refer to Lemma 13.22 for concluding the last assertion.

13.6 The nonperiodic case

| 259

13.6 The nonperiodic case The presence of a noncompact element in a topologically modular group has strong influence on its structure. In fact, as the next result shows, such groups are, in particular, near abelian. Lemma 13.32. Let G be a nonperiodic topologically modular group. Then every compact element generates a normal subgroup of G. The compact elements form an open subgroup of G. In particular, G is near abelian and A := comp(G) is a base group. Proof. Pick any compact element 1 ≠ a in G. Let x be any element with ⟨x⟩ ≅ ℤ discrete. Then the modular law and the fact that ⟨x⟩ ∩ comp(G) = {1} imply ⟨a⟩ = ⟨a⟩ ∨ (⟨x⟩ ∧ comp(G)) = ⟨a, x⟩ ∩ comp(G). Therefore the conjugate xax −1 must belong to ⟨a⟩. Next observe that G \ comp(G) consists of elements, each generating a discrete subgroup isomorphic to ℤ. Then apply Lemma 12.14 with S := comp(G) and deduce that ⟨a⟩ must be normal in G. As a byproduct of this proof we obtain that comp(G) is a characteristic periodic subgroup of G. The group G, being locally compact and totally disconnected, possesses an open compact subgroup, say K. Clearly K is contained in comp(G) showing that comp(G) is an open subgroup of G. For proving that G is near abelian we must prove that comp(G) is abelian and that G/ comp(G) is abelian of torsion-free rank one, i. e., is isomorphic to a subgroup of ℚ. As comp(G) is open and every of its closed subgroups is normal in G, we may select any open compact subgroup, say K. Then G/ comp(G) is topologically isomorphic to the discrete M-group (G/K)/(comp(G)/K) and the latter is of torsion-free rank one, by Iwasawa’s theorem (see Theorem 2.4.11 in [97]). It also follows that comp(G)/K is abelian. Now, as the totally disconnected locally compact group comp(G) is the strong projective limit of all the locally compact abelian groups comp(G)/K with K running through all open compact subgroups of G, it follows that comp(G) is itself abelian. Reasoning similarly as in the last part of the proof, i. e., describing G as a strong projective limit of discrete M-groups, one obtains from Theorem 2.4.11 in [97] that G must be a topologically quasihamiltonian group. We will turn in a moment to further restricting the structure of nonperiodic topologically modular groups. Lemma 13.33. In ℤp let λ := 1 + ps ν for s ≥ 1 and ν ∈ ℤ×p . Fix a prime q ≠ p. Then q−1

e := ∑ λj j=0

is a unit.

260 | 13 Locally compact groups with a modular subgroup lattice Proof. Consider ℤp as a subring of its quotient field ℚp . Then q−1

e = ∑ λj = j=0

λq − 1 (1 + ps ν)q − 1 . = λ−1 ps

Binomial expansion of the numerator gives q

q q (1 + ps ν) − 1 = ∑ ( )psk νsk − 1 k k=0 q

q = ps (ν + ∑ ( )ps(k−1) νsk ). k j=2 Therefore (1 + ps ν)q − 1 = ps ν󸀠 for ν󸀠 being the term in brackets. Observing that ν󸀠 − ν ∈ pℤp shows that ν󸀠 ∈ ℤ×p . Hence e=

(1 + ps ν)q − 1 ν󸀠 = ps ν ν

is a unit, as has been claimed. Lemma 13.34. Let G = ℤp ⋊ ℤ be a semidirect product. Then G cannot be topologically modular. Proof. Fix x ∈ G, a topological generator of ℤp × {1}, and y ∈ G with ⟨y⟩ = {1} × ℤ. We first claim that {1} × ℤ cannot act trivially. Indeed, letting in Example 13.5, q ≠ p any prime and ⟨x⟩ play the role of K and ⟨y⟩ play the role of Z shows that G cannot be topologically modular. Next we claim that y acts on ℤp as (∀a ∈ ⟨x⟩) yay−1 = aλ , where λ is a unit in ℤ×p of the form λ = 1 + ps ν, for s ∈ ℕ and ν ∈ ℤ×p . Certainly conjugation induces an automorphism of ℤp . If the order of this automorphism were finite, say n ∈ ℕ, then G would contain the closed subgroup L := ⟨x, yn ⟩ ≅ ℤp × ℤ. The latter, by our above reasoning, is not topologically modular, and hence G is not topologically modular. The claimed form of λ follows from Proposition 4.17 and recalling the meaning of ℙp from the discussion preceding Lemma 4.14. For finally showing that G is not topologically modular choose a prime q ≠ p and define A := ⟨(xy)q ⟩,

B := ⟨y⟩,

Note that these subgroups of G are all discrete.

and C := ⟨xy⟩.

13.6 The nonperiodic case

| 261

Let us first show that A ∨ B = ⟨y, (xy)q ⟩ is all of G. For proving this it suffices to have x ∈ A ∨ B. Observing xy = xλ , one obtains 2

q−1

(xy)q = xxy xy ⋅ ⋅ ⋅ xy yq = xe yq for e := 1 + λ + ⋅ ⋅ ⋅ + λq−1 . As λ = 1 + ps ν for a unity ν, Lemma 13.33 implies that e is a unity. Hence A ∨ B = ⟨(xy)q , y⟩ = ⟨xe yq , y⟩ = ⟨x e , y⟩ ⊇ ⟨x⟩⟨y⟩ = G. Therefore the terms of the modular identity turn out to be (A ∨ B) ∧ C = G ∧ C = C

and A ∨ (B ∧ C) = ⟨⟨(xy)q ⟩, ⟨y⟩ ∩ ⟨xy⟩⟩ = A.

Thus G is not modular. The following result by Mukhin (see [79, Theorem 2]) constitutes a further restriction. Theorem 13.35. The set tor(G) of torsion elements in a topologically modular group forms an open subgroup and G/ tor(G) is torsion-free of rank 1. Proof. Lemma 13.32 shows that the compact elements of G form a closed periodic abelian subgroup A all of whose closed subgroups are normal in G. Suppose x ∈ A is not a torsion element. If x is a p-element for some prime p and z ∈ ̸ A, then ⟨x, z⟩ = ⟨x⟩ ⋊ ⟨z⟩ and then Lemma 13.34 shows that x must be a torsion element. Now suppose that there is an element x of infinite order in A. Then π := π(⟨x⟩) must be infinite. Pick z ∈ G \ A and note that ⟨x, z⟩ = ⟨x⟩ ⋊ ⟨z⟩. Since ⟨x⟩ = ∏p∈π ⟨xp ⟩ for torsion elements xp , one can replace z by a suitable power so that for p in some finite subset ϕ of π commutes with every xp . Note that x = xϕ xϕ󸀠 and since ⟨x⟩ is procyclic the element xϕ belongs to ⟨x⟩. Replacing x by xp for some p ∈ ϕ shows that there is a commutative subgroup L := ⟨x, z⟩ = ⟨x⟩ × ⟨z⟩. Choose a prime q ≠ p and set A := ⟨(xz)q ⟩, C := ⟨xz⟩, and B := ⟨z⟩. Then (A ∨ B) ∧ C = ⟨xq z q , z⟩ ∧ ⟨(xz)⟩ = ⟨x q , z⟩ ∧ ⟨xz⟩ = ⟨x, z⟩ ∩ ⟨xz⟩ = ⟨xz⟩ = C while A ∨ (B ∧ C) = ⟨(xz)q , ⟨x⟩ ∩ ⟨xz⟩⟩ = ⟨(xz)q , {1}⟩ = ⟨(xz)q ⟩ = A ≠ C. Thus A = comp(G) must be torsion.

262 | 13 Locally compact groups with a modular subgroup lattice Using this result it is easy to provide an example of a topologically quasihamiltonian group which is not topologically modular. We conclude this section by noting that the topologically quasihamiltonian group in Example 12.16 cannot be topologically modular by Lemma 13.34.

14 Strongly topologically quasihamiltonian groups Here we complete the classification of locally compact groups, in which for any closed subgroups X and Y the set XY := {xy : x ∈ X, y ∈ Y} is always a closed subgroup. Thus we answer a question raised in 1984 by Yu. N. Mukhin in the Kourovka Notebook (cf. [66, Problem 9.32]). Call a locally compact group G strongly topologically quasihamiltonian if for every pair of closed subgroups X and Y the set XY := {xy : x ∈ X, y ∈ Y} is a closed subgroup of G. Locally compact abelian groups with this property have been classified by Yu. Mukhin in [78]. The same author asks in [66, Problem 9.32] for a classification of all strongly topologically quasihamiltonian groups. The fact that every strongly topologically quasihamiltonian group is also topologically modular follows immediately from the definitions. We record this fact as follows. Proposition 14.1. Any strongly topologically quasihamiltonian group is topologically modular. However, not every topologically modular group is strongly topologically quasihamiltonian as becomes clear by considering any finite Iwasawa (p, q)-factor; cf. Example 13.15.

14.1 Historical background It was R. Dedekind who in 1897 classified all finite groups G in which every subgroup is normal; see [23]. Such groups are usually termed hamiltonian, and it can be easily shown that the above condition holds in every hamiltonian group. For a discrete group G the condition reduces do XY being a subgroup whenever X and Y are subgroups of G. It was K. Iwasawa who termed these groups quasihamiltonian and provided the classification of all quasihamiltonian groups in [61, 62]. We note that independently G. Zappa obtained classification results in [115, 116]. Corrections of original proofs came from F. Napolitani (see [84]) and M. Suzuki (see [103]). The fact that not every topologically quasihamiltonian group is a strongly topologically quasihamiltonian group, has been observed by Mukhin in [79] and his example, basically Example 13.2, will be reproduced in Lemma 14.5; see the remark thereafter. Mukhin already observed in [81] that any topologically quasihamiltonian group is abelian, provided it has nontrivial connected components. https://doi.org/10.1515/9783110599190-014

264 | 14 Strongly topologically quasihamiltonian groups

14.2 More notation The Frattini subgroup Φ(G) of a compact p-group G is defined as Φ(G) := Gp [G, G], where [G, G] denotes the closure of the commutator subgroup of G and Gp is the closed subgroup of G generated by all pth powers of elements in G. Similarly as in finite p-group theory and as explained in [93], Φ(G) consists of the set of all nongenerators, i. e., all elements in G that can be discarded from any set of topological generators X of G such that X \ Φ(G) still topologically generates the compact p-group G.

14.3 Generalities Now we collect a few simple facts. It is an obstruction to easier proofs that the class of strongly topologically quasihamiltonian groups is not closed under forming projective limits; see Remark 14.6. The good properties of the class are the following ones. Proposition 14.2. The class of strongly topologically quasihamiltonian groups is closed under (a) passing to closed subgroups; and (b) passing to factor groups modulo closed normal subgroups. For its proof we first establish an elementary fact. Lemma 14.3. Let G be a topological group and N a closed normal subgroup. Then, any subgroup S containing N is closed in G if and only if S/N is a closed subgroup of G/N. Proof. Suppose first that S is closed. Since G \ S is open and G/N is equipped with the quotient topology, the set G/N \S/N, being the image of the saturated open subset G\S of G, is itself open in G/N. Hence S/N is indeed a closed subgroup of G/N. For proving the converse, let ϕ denote the canonical projection from G onto G/N. Then, since S/N, by assumption, is closed, its complement G/N \ S/N is open. Because ϕ is continuous and the preimage of G/N \ S/N is open and agrees with G \ S, it follows that S is closed. Proof of Proposition 14.2. Let G be a strongly topologically quasihamiltonian group and L be a closed subgroup. Then the product of any two closed subgroups of L is a closed subgroup of G and hence of L. Thus L is strongly topologically quasihamiltonian. The fact that G/N is strongly topologically quasihamiltonian follows from Lemma 14.3. If a periodic group is the direct product of groups G and H and π(G) ∩ π(H) = 0 it will suffice to ensure that each factor is strongly topologically quasihamiltonian, in order to prove that G × H is strongly topologically quasihamiltonian.

14.4 The abelian case

| 265

Lemma 14.4. If G and H are both periodic strongly topologically quasihamiltonian groups and π(G) ∩ π(H) = 0, then their cartesian product G × H is a strongly topologically quasihamiltonian group. Proof. Put π := π(G) and σ := π(H). For closed subgroups X and Y there is a corresponding decomposition X = Xπ × Xσ ,

Y = Yπ × Yσ .

Then Xπ Yπ and Yσ Yσ are both closed subgroups in respectively G and H by our assumptions. Hence XY = Xπ Yπ × Xσ Yσ is a closed subgroup of G × H.

14.4 The abelian case We begin with some generalities and in subsections we shall deal separately first with the p-case, then with the periodic case, and finally with the nonperiodic case. The following fact has already been observed in [78]; see also Example 13.2. Throughout this section we use additive notation. Lemma 14.5. Let I be a nonempty index set and select for every i ∈ I a prime pi . Set A := ⨁ ℤ(pi ) × ∏ ℤ(pi ). i∈I

i∈I

Then A is strongly topologically quasihamiltonian if and only if I is finite. Proof. Suppose that A is strongly topologically quasihamiltonian. Let ci be a topological generator of the ℤ(pi ) in the profinite factor C := ∏ ℤ(pi ) i∈I

of A and bi for ℤ(pi ) in the discrete factor B := ⨁ ℤ(pi ) i∈I

of A, the infinite direct sum. Then A = B ⊕ C, where C is a compact open subgroup of A. Define X to be the closed subgroup generated by the elements ai := ci + bi and note that X ≅ ⨁i∈I ℤ(pi ) is a discrete subgroup

266 | 14 Strongly topologically quasihamiltonian groups of A. Let Y be the closed subgroup generated by all elements bi . Then, A being strongly topologically quasihamiltonian by assumption, X + Y is a closed subgroup of A and contains all elements ci with i ∈ I. Therefore X + Y must contain the closure of the group generated by all ci , i. e., C. The latter compact group can be a subgroup of the discrete group X + Y only if C is finite. But then I must be finite. Conversely, if I is finite, then A is finite and hence is a strongly topologically quasihamiltonian group. Remark 14.6. Several remarks are in order. (a) Letting, in the preceding lemma, I be an infinite set yields an example of a topologically quasihamiltonian group which is not strongly topologically quasihamiltonian. (b) If I is infinite countable and the primes pi are pairwise different, then A is topologically quasihamiltonian and possesses the discrete subgroups X := B and Y := C with X + Y a nonclosed subgroup. Note that the group B ⊕ C is the projective limit of discrete topologically quasihamiltonian groups. (c) The example in (b) is topologically modular (cf. Theorem 13.31) and, being abelian, is topologically quasihamiltonian, but it is not strongly topologically quasihamiltonian. (d) If D is a divisible group, then D either is discrete or has finite p-rank.

14.4.1 The Abelian p-group case We now turn to refining results of Mukhin in [78]. Lemma 14.7. Let the locally compact abelian p-group contain a finitely generated open subgroup U. Then G is strongly topologically quasihamiltonian. Proof. Passing to a smaller open subgroup of U one can achieve U to be torsion-free. Then tor(G) turns out to be a discrete, hence closed subgroup of G. Let X and Y be closed subgroups of G. Since X ∩ U + Y ∩ U is the sum of compact groups it is closed. Taking Lemma 14.3 into account we may factor X ∩ U + Y ∩ U and thus assume X ∩ U = Y ∩ U = {0}. Then, however, X + Y is a subgroup of tor(G) and is therefore closed. Theorem 14.8. The following statements about a locally compact abelian p-group are equivalent: (a) G is topologically modular, (b) G is strongly topologically quasihamiltonian. Proof. Assume (a) and let U be an open compact subgroup of G. If rankp (U) is finite then G is strongly topologically quasihamiltonian by Lemma 14.7. We assume from now on that the p-rank of U is infinite. Let X and Y be closed subgroups of G. Taking

14.4 The abelian case

| 267

Lemma 14.3 into account, we may factor X ∩ U and Y ∩ U and henceforth assume that X and Y are torsion subgroups of G. Taking Theorem 13.11 (b) into account we find that the closed torsion subgroups X and Y both must have finite p-rank. The closure of X +Y has finite p-rank by Lemma 3.91. Therefore Theorem 3.97 implies that X + Y, having a dense torsion subgroup, must be torsion itself. Hence X + Y is closed. Thus (b) holds. Assuming (b), we have (A ∨ B) ∧ C = A ∨ (B ∧ C) for closed subgroups A ⊆ C and B follows from the containments (A ∨ B) ∧ C = (A + B) ∩ C ⊆ A + (B ∩ C) = A ∨ (B ∧ C), A ∨ (B ∧ C) = A + (B ∩ C) ⊆ (A + B) ∩ C = (A ∨ B) ∧ C. Theorem 14.9. The following statements for a locally compact abelian p-group G are equivalent: (a) G is topologically modular, (b) for U an open compact subgroup exclusively one of the following holds: (b.1) U has finite p-rank; then tor(G) is discrete and G/ tor(G) has finite p-rank, (b.2) U has infinite p-rank; then div(G) has finite p-rank and G/ div(G) is compact, (c) G is strongly topologically quasihamiltonian. Proof. Let (a) be true. Then from Theorem 13.11 (b) follows. Assume (b.1). This condition holds for closed subgroups and factor groups. Let X and Y be any closed subgroups. Then, letting UX and UY be open compact subgroups of respectively X and Y, one may pass to the factor group H := G/(UX + UY ) and hence assume that X and Y are torsion. Since in H the torsion subgroup is discrete we conclude that (X + Y)/(UX + UY ) is discrete and hence closed. Assume (b.2). Let X and Y be any closed subgroups and, similarly as before, factor UX + UY in G. Then, in H := G/(UX + UY ) the images of respectively X and Y are discrete torsion subgroups. Since G/ div(G) is compact it follows that (X +Y)/(UX +UY ) is compact and hence finite. Thus (b) implies (c). The fact that (c) implies (a) has been shown in Theorem 14.8. More can be said if G is torsion. Corollary 14.10. Let G be a locally compact abelian nondiscrete torsion p-group. Then G is topologically modular if and only if the maximal divisible subgroup D := div(G) has finite p-rank and there is a compact open, and hence reduced, subgroup R such that G =R⊕D algebraically and topologically.

268 | 14 Strongly topologically quasihamiltonian groups Proof. We discuss the cases in Theorem 13.11. If the premise in (a) holds, then, since U is a compact torsion group having finite exponent and finite p-rank, it is finite. Then G would have to be discrete, contrary to our assumptions. Thus G satisfies premise (b). Therefore the p-rank of G/U is finite and D is a finite p-rank divisible subgroup—hence D is a discrete subgroup of G. Passing to a smaller open compact subgroup if necessary we can assume D ∩ U = {0}. Using a result of R. Baer (see [32, Theorem 22.2], one can find a reduced subgroup R of G containing the reduced subgroup U with G = D ⊕ R. Necessarily R is open in G. Since G/D is compact conclude that (R + D)/D is compact. An application of the open mapping theorem (see [54, Exercise EA1.21. p. 704]) shows that the latter group is algebraically and topologically isomorphic to R. Hence R is itself compact. Immediate consequences are refined structure results for reduced, for torsion, and for divisible locally compact abelian p-groups. Corollary 14.11. Let G be a reduced locally compact abelian torsion p-group. To be strongly topologically quasihamiltonian it is necessary and sufficient that G is either discrete or compact. Corollary 14.12. Let G be a locally compact abelian torsion strongly topologically quasihamiltonian p-group. Then either G is discrete or its maximal divisible subgroup D has finite p-rank—even when considered as a discrete subgroup. In particular D is a closed subgroup of G. For the divisible p-group case we have a helpful observation. Proposition 14.13. Let G be a strongly topologically quasihamiltonian group having discrete maximal divisible subgroup D := div(G). Then G = D ⊕ R for a reduced subgroup algebraically and topologically. Proof. Since D is discrete there is an open compact subgroup U of G intersecting D trivially. The discrete group G/U allows a direct decomposition G/U = ((D + U)/U) ⊕ R/U, for R an open subgroup of G with R/U reduced. Observe that G = D + R and D ∩ R ⊆ D ∩ ((D + U) ∩ R) ⊆ D ∩ U = {0}. Hence G is algebraically the direct sum G = D ⊕ R. Since D ⊕ U is an open and hence closed subgroup of G we find that G = D ⊕ R also topologically. Two additional consequences may be concluded and only the second one needs a proof.

14.4 The abelian case

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Corollary 14.14. Let A be a locally compact abelian torsion p-group, not discrete. Then A is strongly topologically quasihamiltonian if and only if A can be decomposed into a direct sum of closed subgroups A = D ⊕ V, where D is a discrete divisible torsion group of finite p-rank and V is a compact subgroup of A. Corollary 14.15. Let G be a locally compact abelian strongly topologically quasihamiltonian nondiscrete torsion p-group and D its maximal divisible subgroup D. Then D is closed and, for C a compact subgroup of G, D + C/C is the maximal divisible subgroup of G/C. Proof. By Corollary 14.12 the maximal discrete subgroup D of G turns out to be a discrete and finite p-rank subgroup of G. Corollary 14.14 implies the existence of a compact subgroup V of G and an algebraic and topological direct decomposition G = D ⊕ V. Factoring the finite subgroup D∩C implies that G/D∩C has maximal divisible subgroup D/D ∩ C. By the finiteness of D ∩ C the factor group G/D ∩ C is a nondiscrete strongly topologically quasihamiltonian torsion p-group. Thus, in the sequel, assume D ∩ C = {0}. Then there exists an open compact subgroup U containing C with D ∩ U = {0}. Using a result of Baer (see [32, Theorem 21.2]), one can find a reduced subgroup R of G containing U such that G = D ⊕ R. Since R is reduced and not discrete it is compact as a consequence of Corollary 14.11. Therefore G/C ≅ D ⊕ R/C, where R/C is compact and hence reduced. Thus D + C/C ≅ D is indeed the maximal divisible subgroup of G/C. Every compact or discrete abelian p-group clearly is strongly topologically quasihamiltonian. Therefore we first concentrate on groups neither compact nor discrete. Proposition 14.16. The following statements about a locally compact abelian reduced p-group G, neither discrete nor compact, are equivalent: (a) G is strongly topologically quasihamiltonian, (b) G contains an open finitely generated subgroup.

270 | 14 Strongly topologically quasihamiltonian groups Proof. Suppose (a). Then by Theorem 14.8 G is topologically modular and Theorem 13.11 (a) implies (b) of our proposition. The fact that (b) implies (a) is an immediate consequence of Lemma 14.7. Remark 14.17. Theorem 3.17 discusses a reduced locally compact abelian p-group which is strongly topologically quasihamiltonian and is neither compact nor discrete. Proposition 14.18. A divisible locally compact abelian p-group G is strongly topologically quasihamiltonian if and only if, for some set I and nonnegative integer m, G ≅ ℤ(p∞ )

(I)

⊕ ℚm p

algebraically and topologically. Proof. Assume first that G is strongly topologically quasihamiltonian. If G is discrete it has the described structure for m = 0. Thus we may assume G not to be discrete. Fix an open compact subgroup U of G. Then G/pU is a torsion strongly topologically quasihamiltonian group and Corollary 14.14 implies that G/pU either is discrete or has finite p-rank. In either case U/pU is finite and hence U has finite p-rank. Therefore, by Theorem 3.97, U ≅ F ⊕ ℤm p for some nonnegative integer m and a finite subgroup F of G. Choosing the open compact subgroup U small enough, we can arrange U ∩ tor(G) = {0}. Hence the subgroup tor(G) is discrete and divisible and is therefore, as a consequence of Lemma 3.13, topologically and algebraically a direct summand of G intersecting U trivially. Then G = tor(G) ⊕ DU , with DU ≅ ℚm p a divisible hull of U. Since tor(G) is discrete and divisible there is a set I with tor(G) ≅ ℤ(p∞ ) . (I)

Conversely, suppose G ≅ ℤ(p∞ )

(I)

⊕ ℚm p

for I some set and m ≥ 0. Then, taking Lemma 14.7 into account, it suffices to remark that for any open compact subgroup U contained in ℚm p , the factor group G/U is discrete. Hence G is strongly topologically quasihamiltonian.

14.4 The abelian case

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Lemma 14.19. Let G be a strongly topologically quasihamiltonian abelian p-group with a finitely generated open subgroup U. Then the maximal divisible subgroup D is closed and G algebraically and topologically decomposes G = D ⊕ R, for R a closed reduced subgroup of G. Proof. As U is finitely generated we may replace it by an open subgroup in order to achieve that U is also torsion-free. Then certainly D ∩ U is finitely generated and hence by Proposition 3.76 D is closed and D∩U is finitely generated. Again passing to an open subgroup of U, if necessary, it turns out that D∩U can be made a direct summand of U. Therefore the same proposition implies the existence of a closed reduced subgroup R of G so that algebraically and topologically G = D ⊕ R. 14.4.2 Classifying periodic abelian groups As a main result we present a full classification of the periodic strongly topologically quasihamiltonian groups in Theorem 14.22. The fact that inductively monothetic groups are always strongly topologically quasihamiltonian could be concluded from Mukhin’s characterization of abelian strongly topologically quasihamiltonian groups (see [78]); a very elementary proof follows for the convenience of the reader. Proposition 14.20. Every periodic inductively monothetic group H is strongly topologically quasihamiltonian. Proof. Assume first that H is a p-group. Then, as a consequence of Lemma 5.5, H is isomorphic either to the additive group of the p-adic field ℚp , to the additive group of the p-adic integers ℤp , to a finite cyclic p-group, or to Prüfer’s group ℤ(p∞ ). Then, for X and Y any closed subgroups of H, either X ⊆ Y or Y ⊆ X must hold. But then XY agrees with one of the closed subgroups Y or X and is hence a closed subgroup of H. Let H now be arbitrary and consider closed subgroups X and Y. Put π := {p ∈ π(H) : Xp ⊆ Yp }. Then X = Xπ × Xπ 󸀠 and Y = Yπ × Yπ 󸀠 . For proving XY to be a closed subgroup of H, in light of Lemma 14.4, it suffices to prove closedness of the two subgroups Xπ Yπ and Xπ 󸀠 Yπ 󸀠 . In the first case the group in question agrees with the closed subgroup Yπ and in the second one with the closed subgroup Xπ 󸀠 . Hence XY is a closed subgroup. Thus H is strongly topologically quasihamiltonian. We first present a complete description of all torsion locally compact abelian strongly topologically quasihamiltonian groups.

272 | 14 Strongly topologically quasihamiltonian groups Theorem 14.21. A locally compact abelian torsion group A is strongly topologically quasihamiltonian if and only if there is a partition π(A) = κ ∪ ϕ and all of the following holds: (i) The set of primes ϕ is finite and Aϕ = Dϕ ⊕ Vϕ for Dϕ divisible and Vϕ compact. For every p ∈ ϕ the p-primary subgroup Dp of Dϕ has finite p-rank. (ii) The subgroup Aκ is a discrete subgroup of A. Proof. Assume first that A is strongly topologically quasihamiltonian. Select an open compact subgroup, say U, of A. Since A is torsion, U is a compact abelian torsion group, and therefore the set ϕ := π(U) must be finite. Put κ := π(A) \ ϕ. From Aκ ∩ U = {0} it follows that Aκ is a discrete subgroup of A, proving (ii). Next observe that Aϕ = ⨁ Ap p∈ϕ

is a direct sum since ϕ is finite. Corollary 14.14 implies that for each p in ϕ there is a decomposition Ap = Dp ⊕ Vp , with Dp a divisible finite p-rank subgroup of A and Vp compact. Thus there is a decomposition Aϕ = Dϕ ⊕ Vϕ , where Dϕ := ⨁p∈ϕ Dp is a discrete divisible subgroup and Vϕ := ⨁p∈ϕ Vp is compact. Thus also (i) is established. For proving the converse, assume (i) and (ii) to hold. Apply Lemma 14.4 to Aκ and the finitely many factors Ap for p ∈ ϕ. Then Aκ , being discrete, is strongly topologically quasihamiltonian and by Corollary 14.14 so is Ap for every p in ϕ. We can now complete Mukhin’s classification of abelian strongly topologically quasihamiltonian groups. During the proof of the next theorem we let for a profinite abelian group U the Frattini subgroup Φ(U) be the intersection of all maximal open subgroups of U. As pointed out in [93] this group is for G = ∏p Ap of the form Φ(A) = ∏ pAp . p

14.4 The abelian case

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Theorem 14.22. For a locally compact abelian periodic group A and open compact subgroup U the following statements are equivalent: (A) A is strongly topologically quasihamiltonian, (B) there is a partition of π(A) into disjoint subsets κ, ϕ, and μ := π(A) \ (κ ∪ ϕ) and all of the following holds: (i) κ = {p ∈ π(A) : Ap ∩ U = {0}} and Aκ is a discrete subgroup of A, (ii) ϕ = {p ∈ π(A) \ κ : rankp (Ap ) ≥ 2}, the set ϕ is finite, and for all p ∈ ϕ the p-Sylow subgroup Ap is strongly topologically quasihamiltonian, (iii) Aμ is inductively monothetic, (iv) A = Aκ × Aϕ × Aμ topologically and algebraically. Proof. Assume (A). Then certainly Aκ ∩ U = {0} and hence Aκ is a discrete κ-Sylow subgroup of A. It follows that A = Aκ × Aκ󸀠 , algebraically and topologically. Thus (i) is established. For establishing (ii) and (iii) we may restrict ourselves to the case κ = 0, i. e., Aκ = {0} from now on. Since, by Proposition 14.2, every subgroup and quotient of a strongly topologically quasihamiltonian group again is a strongly topologically quasihamiltonian group, passing to subgroups of quotients of Aϕ renders strongly topologically quasihamiltonian groups. For proving that ϕ must be finite we may factor Φ(Uϕ ) = ∏p∈ϕ pUp and achieve that Up has exponent p only. By using Corollary 5.4 one can find for every p ∈ ϕ elements ap ∈ Ap \ Up of order at most p2 and 0 ≠ bp ∈ Up with ⟨ap ⟩ ∩ ⟨bp ⟩ = {0}. The closed subgroup L := ⟨ap , bp : p ∈ ϕ⟩ of Aϕ is still strongly topologically quasihamiltonian. Observe that U ∩ L

=

⟨pap , bp : p ∈ ϕ⟩ and, after factoring in it the closed subgroup N generated by all elements of the form pap , we find an algebraic and topological isomorphism L/N ≅ ⨁ ℤ(p) ⊕ ∏ ℤ(p). p∈ϕ

p∈ϕ

Since L/N is strongly topologically quasihamiltonian deduce from Lemma 14.5 that ϕ must be finite. Hence (ii) holds. (iii) This is an immediate consequence of the fact that p ∈ μ if and only if rankp (Ap ) = 1 if and only if Ap is inductively monothetic; see Proposition 5.2 and Definition 5.9. Finally, A certainly is the cartesian product of the Sylow subgroups Aκ , Aϕ , and Aμ . Hence (B) holds.

274 | 14 Strongly topologically quasihamiltonian groups Assume now (B). In light of Lemma 14.4 it will suffice to prove that Aκ , Aϕ , and Aμ are strongly topologically quasihamiltonian. For Aκ this is obvious since Aκ is discrete. For every p in the finite set ϕ we know that Ap is strongly topologically quasihamiltonian. Thus applying Lemma 14.4 to the finite product Aϕ = ∏ Ap p∈ϕ

shows that Aϕ is strongly topologically quasihamiltonian. Finally observe that Aμ has p-rank 1 p-Sylow subgroups for all p ∈ μ. Hence Aμ is standard inductively monothetic in the sense of Definition 5.9. Applying Proposition 14.20 shows that Aμ is strongly topologically quasihamiltonian. Remark 14.23. In Corollary 14.14 we could have tried to invoke a result by Sahleh and Alijani [94, Theorem 2.2] for finding the direct decomposition A = D ⊕ V. However, our proof is direct and does not touch the discussion of the validity of Theorem 1 in [34].

14.5 The nonabelian situation In separate subsections we shall deal with the periodic and with the nonperiodic case.

14.5.1 The periodic case A classification of the periodic strongly topologically quasihamiltonian groups will be given by means of Theorem 14.28 and Proposition 14.31 in terms of a strongly topologically quasihamiltonian base group A. In conjunction with Theorem 14.22 and the results in Section 14.4.1 on locally compact abelian p-groups the classification of strongly topologically quasihamiltonian groups is complete. In particular, recall that a locally compact abelian p-group is strongly topologically quasihamiltonian iff it is topologically modular (cf. Theorem 14.9). Proposition 14.24. For a compactly ruled p-group G the following statements are equivalent: (a) G is topologically modular, (b) G is strongly topologically quasihamiltonian. Proof. In light of Proposition 14.1 it suffices to prove that (a) implies (b). Thus assume G to be a topologically modular group. Then any closed subgroup is a topologically modular group as well. Lemma 13.20 implies that G is near abelian. If G is abelian we are done by Theorem 14.9. Let us thus assume G not to be abelian. Then by the very definition of near abelian (see Definition 6.1) there is a base subgroup A and a scaling

14.5 The nonabelian situation

| 275

subgroup H of G such that G = AH, and according to Theorem 7.9 the subgroup H is a procyclic p-group. Let now X and Y be closed subgroups of G. Making use of Proposition 7.39 one can devise closed subgroups AX and AY of A and closed procyclic subgroups HX and HY of G such that X = AX HX and Y = AY HY . Note that by the normality of AY and AX in G XY = AX HX AY HY = AX AY HX HY

and YX = AY AX HY HX = AX AY HY HX

and by Theorem 14.9 A is strongly topologically quasihamiltonian, so that AX AY = AY AX is a closed subgroup of A. We need to prove that XY = YX

(*)

and we claim that it will suffice to ensure that L := HX HY = HY HX is a compact p-group of the form L = AL HL with AL a closed subgroup of A and HL a procyclic p-group. Indeed, if this is true, then XY = AX AY HX HY = AX AY AL HL = AY AX HY HX = YX is a closed subgroup of G since AX AY AL is a closed normal subgroup of G and HL is a procyclic p-group. Thus we can assume X = HX and Y = HY and we note that both of them are procyclic p-groups (see Theorem 7.9). Since G = ⟨X, Y⟩ is periodic it is a compact topologically modular p-group. It is a consequence of [93, Proposition 2.1.4] that for 𝒩 a filter basis of open normal subgroups of G XY = ⋂ XYN N∈𝒩

and YX = ⋂ YXN, N∈𝒩

so that it will suffice to ensure (XN/N)(YN/N) = (YN/N)(XN/N) for all open normal subgroups N ∈ 𝒩 . Thus G = ⟨X, Y⟩ can be assumed to be finite. Since G/A is a finite cyclic group we may, without losing generality, assume that AY ⊆ AX and hence, if X = ⟨b⟩, there are a ∈ A and λ ∈ ℤ with Y = ⟨abλ ⟩. Let G have minimal order such that equation (*) fails to hold and pick s in the socle of socle(⟨a⟩). Note that a ∈ YX and b−λ abλ ∈ XY imply s ∈ XY ∩ YX,

276 | 14 Strongly topologically quasihamiltonian groups so that factoring the normal subgroup ⟨s⟩ implies XY = XY⟨s⟩ = YX⟨s⟩ = YX, a contradiction. In a similar vein we can describe near abelian strongly topologically quasihamiltonian groups. Proposition 14.25. Let G = AH be a periodic near abelian group with base A and procyclic scaling group H. The following statements are equivalent: (A) The group G is topologically quasihamiltonian and the base A is strongly topologically quasihamiltonian. (B) The group G is strongly topologically quasihamiltonian. Proof. Assume first (A) to hold. Thus we assume that G is topologically quasihamiltonian and A is strongly topologically quasihamiltonian. Let X and Y be arbitrary closed subgroups of G. Then, letting AX := X ∩ A and AY := Y ∩ A, Lemma 7.39 implies the existence of respective procyclic scaling subgroups HX and HY such that X = AX HX ,

Y = AY HY .

Observe that the normality of AX and AY in G implies the chain of equalities XY = AX HX AY HY = AX AY HX HY .

(14.1)

The subgroup B := AX AY is closed since A is supposed to be strongly topologically quasihamiltonian. Since HX and HY are procyclic and G is topologically quasihamiltonian we may deduce that C := HX HY = HX HY = HY HX = HY HX is a compact subgroup of G. Then, setting AC := C ∩ A and invoking Lemma 7.39 once more, one can find a procyclic scaling group HC with C = AC HC . Therefore equation (14.1) implies XY = BAC HC . Since A is strongly topologically quasihamiltonian, the subset BAC is a closed abelian subgroup of A and since HC is compact, it follows that XY is a closed subgroup of G. Hence G is strongly topologically quasihamiltonian, i. e., (B) is established. Assuming (B) to hold, it follows from the very definition of strongly topologically quasihamiltonian that G is topologically quasihamiltonian. Then Proposition 14.2 implies that A is strongly topologically quasihamiltonian. Thus (A) is verified.

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Corollary 14.26. Let G be periodic topologically quasihamiltonian and π(G) be finite. Suppose A is a base and H a Π-procyclic scaling subgroup. Then G is strongly topologically quasihamiltonian if and only if A is strongly topologically quasihamiltonian. Proof. According to Theorem 12.11, and because π(G) is finite, G = ∏ Gp p∈π(G)

is the finite direct product of its p-Sylow subgroups Gp . Therefore H = ∏ Hp p∈π(G)

is compact. Now the result follows from Proposition 14.25. During the proof of the next result we make use of the notion of the Frattini subgroup Φ(G) of a pronilpotent group G; Φ(G) is the intersection of all open maximal subgroups of G and, as explained in [93], for the pronilpotent group G being the cartesian product G = ∏p Gp it takes the form Φ(G) = ∏p Φ(Gp ). Finally, as noted ibidem, Φ(Gp ) = Gpp G󸀠 . Lemma 14.27. Let G = AH be a topologically quasihamiltonian group with compact base A and Π-procyclic scaling group H. Then G is a strongly topologically quasihamiltonian group if and only if there are pairwise disjoint sets of primes ϕ, γ, α, and σ forming a partition of π(G), and all of the following conditions hold: (a) G is the cartesian product G = Gϕ × Gγ × Gα × Gσ , (b) The set of primes ϕ is finite, (c) Gγ is compact, (d) Gα = Aα , (e) Gσ = Hσ . Proof. Assume first that G is strongly topologically quasihamiltonian and let us prove that (a)–(e) must hold for a suitable partition of π(G). As π(G) will be the disjoint union of sets of primes ϕ, γ, α, and σ, Lemma 14.4 implies the decomposition of G as a direct product as in (a). We are now going to explicitly construct the sets in the partition of π(G) and establish the properties (b)–(e) of the corresponding Sylow subgroups of G. Since G is a topologically quasihamiltonian group, by Theorem 12.11 there is a decomposition of G as a local direct product loc

G = ∏(Gp : Up ). p

278 | 14 Strongly topologically quasihamiltonian groups As A is compact, there is no loss of generality to assume U to contain A and some compact open subgroup CH of H. Set γ := {p ∈ π(G) : Hp ⊆ CH }, i. e., p is in γ if and only if Hp agrees with the p-Sylow subgroup of CH . Then Hγ is contained in CH and thus compact. Hence Gγ is compact. Decompose G = Gγ × Gγ 󸀠 , where γ 󸀠 := π(G) \ γ. Then we are left with proving (b), (d), and (e) for Gγ󸀠 . Thus we allow ourselves to proceed in the sequel under the additional assumption that γ = 0. Put α := {p : Gp = Ap } and, similarly as dealing with γ, we may split the direct factor Gα satisfying (d). Thus from now on we may assume in addition to γ = 0 that also α = 0, in particular, that Hp \ U ≠ 0 holds for all p ∈ π(G). Next put σ := {p ∈ π(G) : Gp = Hp } and split Gσ = Hσ as a direct factor. Thus (e) holds for Gσ by the very definition of Gσ . Thus from now on we may assume, in addition to all assumptions above, that σ = 0. Then one has Hp \ U ≠ 0 for all p ∈ π(H) and π(A) = π(H). Claim: For all p in π(G) one has Hp ⊈ Ap (Hp ∩ U). Suppose, by contradiction, the claim to be false. Then there is p ∈ π(G) with Hp ⊆ Ap (Hp ∩ U). Since Hp is a procyclic compact p-group with Hp \ Hp ∩ U ≠ 0, the intersection Hp ∩ U must be a subgroup of the Frattini subgroup Φ(Hp ) of Hp . Therefore, as Φ(Hp ) ⊆ Φ(Gp ), our assumption Hp ⊆ Ap (Hp ∩ U) would imply Hp ⊆ Ap Φ(Gp ). Then the series of containments Gp = Ap Hp ⊆ Ap (Hp ∩ U) ⊆ Ap Φ(Gp ) ⊆ Gp follows. But then, by the properties of the Frattini subgroup, Gp must agree with Ap showing that p ∈ α—contradicting to α being the empty set. The claim holds. Let us resume the proof of the lemma. It follows from the claim that π(G) ⊆ π(H) and thus π(G) = π(H). Moreover, as Gp = Ap Hp and Ap and Hp are compact, so is Gp . Furthermore, as for every p ∈ π(G) the Frattini subgroup Φ(Gp ) contains Φ(Ap )(Hp ∩U), it follows that π(G/Φ(A)(H ∩ U)) = π(G). Factoring the Frattini subgroup Φ(A) := ∏p∈π(A) Ap of A yields the quotient group A/Φ(A) on which, as a consequence of Theorems 12.5 and 12.11, H acts trivially by

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conjugation. The homomorphism ψ : H → D := HΦ(A)/Φ(A)(H ∩ U) has the compact kernel H ∩ U since A ⊆ U. Therefore algebraically and topologically D ≅ H/H ∩ U is a discrete inductively monothetic subgroup of G/Φ(A)(H ∩ U). Moreover, K := A(H ∩ U)/Φ(A)(H ∩ U) is a compact subgroup and K ∩ D = {1}. Thus, the factor group G/Φ(A)(H ∩ U) is isomorphic to K × D, where K is a compact abelian group and D a discrete inductively monothetic group, and π(K) = π(D) = π(G). Moreover, for every p ∈ π(G) the p-Sylow subgroup of K is abelian of exponent p and the p-Sylow subgroup of D is a finite cyclic p-group. For every p ∈ π(G) pick cp ∈ K and bp of order p in the finite cyclic group Dp . Then the closed subgroup of K generated by the elements cp and bp is isomorphic to ∏p∈π(G) ⟨cp ⟩ × ⨁p∈π(G) ⟨bp ⟩ and is strongly topologically quasihamiltonian. Therefore Lemma 14.5 implies the finiteness of π(G). In order to be coherent with notation, put ϕ := ϕ(G). Thus we finished establishing the properties (a)–(e). Conversely, for proving that the conditions (a)–(e) imply that G is strongly topologically quasihamiltonian, remark first that Gϕ is a finite product of its compact Sylow subgroups and hence is strongly topologically quasihamiltonian by Corollary 14.26. Moreover, Gα = Aα certainly is strongly topologically quasihamiltonian, in light of Proposition 14.2. Since Gγ is compact it automatically is a strongly topologically quasihamiltonian group. Finally note that Gσ = Hσ is inductively monothetic and therefore, by Proposition 14.20, it is strongly topologically quasihamiltonian. By Lemma 14.4 the cartesian product of the groups Gϕ , Gγ , Gα , and Gσ is therefore a strongly topologically quasihamiltonian group. We are ready for classifying strongly topologically quasihamiltonian groups that are split extensions of a base and a scaling subgroup. Theorem 14.28. Let the topologically quasihamiltonian group G be a semidirect product G = A ⋊ H of a strongly topologically quasihamiltonian base A and Π-procyclic scaling subgroup H. Then G is strongly topologically quasihamiltonian if and only if π(G) admits a partition π(G) = ϕ ∪ γ ∪ σ ∪ δ and the corresponding Sylow subgroups enjoy the following properties: (a) ϕ is finite, (b) Hγ is compact, (c) Gσ = Hσ , i. e., Aσ = {1}, (d) Gδ is discrete, (e) G = Gϕ × Gγ × Gσ × Gδ . Proof. Suppose first that G = A ⋊ H is strongly topologically quasihamiltonian. Fix an open compact subgroup C of A. Then the closed subgroup L := CH ≅ C ⋊ H

280 | 14 Strongly topologically quasihamiltonian groups is a strongly topologically quasihamiltonian group. As C is a compact base of L, Lemma 14.27 shows the existence of a partition π(L) = ϕL ∪ γL ∪ αL ∪ σL with ϕL finite, HγL compact, LαL contained in AαL , and LσL agreeing with HσL . The sets of the partition of π(L) will be modified in order to yield the desired partition of π(G). First set γ := γL ∪ αL and note that Hγ = HγL × HαL is compact since HαL = {1}. Next consider GσL = AσL ⋊ HσL and observe that AσL ∩ C = CσL = {1} by the very definition of σL (cf. Lemma 14.27). Therefore AσL is a discrete subgroup of G and hence one can select, for every prime p in σL , an element bp of order p in Ap . These elements generate a discrete subgroup, say S, of A with π(S) = σL . Then H acts trivially by conjugation on S and therefore the subgroup T := ⟨S, H⟩ = S × HσL is a closed subgroup of G and hence strongly topologically quasihamiltonian. Fix an open compact subgroup K of HσL , put κ := π(K), and note the topological isomorphism T/Φ(K) ≅ S × K/Φ(K). Since κ is a subset of σL , Lemma 14.5 implies the finiteness of κ. We let ϕ be the union of ϕL and κ and observe that ϕ is finite, as desired. Finally define δ := σL \κ and observe that Hδ ∩ K = {1} implies that Hδ is discrete. Now Gδ = Aδ ⋊ Hδ is discrete. Item (e) is a consequence of Theorem 12.11. For proving the converse, assume that G is topologically quasihamiltonian, that A is strongly topologically quasihamiltonian, and that π(G) admits the above partition with Sylow subgroups enjoying the properties listed. It will suffice to invoke Lemma 14.4 and prove that each of the subgroups Gϕ , Gγ , Gδ are strongly topologically quasihamiltonian. For Gϕ = ∏p∈ϕ Gp it suffices to remark that it is a finite direct product of the strongly topologically quasihamiltonian groups Gp . Then Corollary 14.26 implies Gϕ to be strongly topologically quasihamiltonian. The γ-Sylow subgroup Gγ = Aγ ⋊ Hγ is strongly topologically quasihamiltonian by Proposition 14.25. The inductively monothetic group Gσ = Hσ is strongly topologically quasihamiltonian by Proposition 14.20. For the discrete group Gδ the properties topologically quasihamiltonian and strongly topologically quasihamiltonian agree. Hence Gδ is strongly topologically quasihamiltonian. Lemma 14.29. Let p be an odd prime and G be a finite p-group. Suppose that U is a cyclic central subgroup of G and U ∩ ⟨g⟩ is not trivial for any nontrivial element g in G. Then G is cyclic.

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Proof. Let S be any subgroup of G of order p, generated by some element g. Then our assumption ⟨g⟩ ∩ U ≠ {1} forces S to coincide with the unique minimal subgroup of U. Therefore G has a unique subgroup of order p and the result follows from [60, p. 310, Satz 8.2]. Lemma 14.30. Let G be a periodic topologically quasihamiltonian group having a strongly topologically quasihamiltonian base A and Π-procyclic scaling subgroup H. Suppose that U is an open compact subgroup of G contained in A ∩ H. For any closed subgroup X of A set θ(X) := {p ∈ π(X) ∩ π(U) : ∃1 ≠ x ∈ Xp such that ⟨x⟩ ∩ U = {1}}. Then θ(X) is finite. Proof. Pick any p ∈ θ(X). Then 1 ≠ (A ∩ H)p because 1 ≠ Up is contained there, and therefore Corollary 7.10 implies that Ap has finite exponent, and so we have Xp and Up , both being contained in Ap . Thus there is an element xp in X of order p such that ⟨xp ⟩ is isomorphic to ℤ(p) and intersects U trivially. Then, selecting for every p ∈ θ(X) an element up of order p in Up , it follows that B := ⟨up : p ∈ θ(X)⟩ and C := ⟨xp : p ∈ θ(X)⟩ give rise to a subgroup C × B ≅ ( ⨁ ℤ(p)) × ( ∏ ℤ(p)). p∈θ(X)

p∈θ(X)

The latter group is closed in the strongly topologically quasihamiltonian subgroup A of G and hence strongly topologically quasihamiltonian. Therefore Lemma 14.5 implies the finiteness of θ(X). When G is not the semidirect product of a base subgroup A and a scaling group H, a reduction statement can be provided. Proposition 14.31. Let G be a periodic topologically quasihamiltonian group with a strongly topologically quasihamiltonian base A and scaling subgroup H. If G/A ∩ H is strongly topologically quasihamiltonian, then so is G. Proof. Consider first the factor group G/A ∩ H and let, in accordance with Theorem 14.28, π(G/A ∩ H) = ϕ ∪ γ ∪ α ∪ σ ∪ δ be a partition such that: (a) ϕ is finite, (b) (H/H ∩ A)γ is compact, (c) (G/H ∩ A)α is contained in A/H ∩ A, (d) (G/H ∩ A)σ is contained in H/H ∩ A, (e) (G/H ∩ A)δ is discrete. We let ϕ contain the prime 2 if it occurs in π(G/A ∩ H).

282 | 14 Strongly topologically quasihamiltonian groups As Theorem 7.15 guarantees, for any subset of primes ω of π(G), a topological isomorphism (G/A ∩ H)ω ≅ Gω /Hω ∩ Aω , it will suffice to prove that Gω is strongly topologically quasihamiltonian, where ω is one of the sets of primes in (a)–(e), and then to invoke Lemma 14.4, in order to achieve the desired result, namely that the finite direct product G = Gϕ × Gγ × Gα × Gσ × Gδ is strongly topologically quasihamiltonian. Thus we individually discuss the cases (a)–(e). (a) If ϕ is finite, then Gϕ is strongly topologically quasihamiltonian as a consequence of Corollary 14.26. (b) By definition Hγ ∩ Aγ is cocompact in Hγ and, according to Lemma 5.25, there is a partition π(Hγ ) = ϕH ∪ γH ∪ κH , such that ϕH is finite, HγH is compact, and HκH agrees with (A ∩ H)κH . Then the subgroup GϕH is strongly topologically quasihamiltonian by Corollary 14.26, GγH has compact scaling subgroup HγH so that it is strongly topologically quasihamiltonian by Proposition 14.25, and, finally, HκH agreeing with (A ∩ H)κH implies that GκH is a closed subgroup of A and hence strongly topologically quasihamiltonian. (c) Now Gα = Aα Hα agrees with Aα , since Hα = {1}, and hence is strongly topologically quasihamiltonian as a closed subgroup of the strongly topologically quasihamiltonian group A. (d) One has Gσ = Hσ and thus Gσ is inductively monothetic and hence strongly topologically quasihamiltonian by Proposition 14.20. (e) Since A/H ∩ A is discrete, there is an open compact subgroup U of A contained entirely in H ∩ A. Recall next that for every p ∈ δ the p-Sylow subgroup Gp = Ap Hp is strongly topologically quasihamiltonian by Corollary 14.26. Lemma 14.30, for X := Gδ , implies that for only finitely many primes p ∈ δ, there is a cyclic subgroup ⟨xp ⟩ of order p, not contained in Up . Let this finite set be denoted by θ and observe that Gθ is strongly topologically quasihamiltonian by Corollary 14.26. Thus, we may replace δ by δ \ θ and assume, during the remainder of the proof, that for all p in δ and every nontrivial p-element x the subgroup Up is contained in ⟨x⟩. Since the prime 2, by construction, belongs to ϕ, all primes p in δ are odd. Then it follows from Lemma 14.29 that Gp is a finite cyclic group, i. e., Gδ is a Π-procyclic inductively monothetic group. Replace σ by σ ∪ δ, as Gδ may now be viewed as the scaling subgroup for the trivial base Aδ .

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14.5.2 The nonperiodic case We deal here with nonperiodic locally compact groups. First we turn to the abelian case and complete our findings about locally compact abelian topologically modular groups in Section 13.2.2. Theorem 14.32. Let G be a totally disconnected locally compact abelian group—neither discrete nor periodic. Then the following statements are equivalent: (a) G is topologically modular, (b) (b.1) for some finite set of primes ϕ and its complement κ = π(tor(G)) \ ϕ, we have tor(G) = tor(G)ϕ ⊕ tor(G)κ , where tor(G)ϕ = div(tor(G))ϕ ⊕ E for E a compact subgroup of finite exponent with π(E) ⊆ ϕ and div(tor(G))ϕ discrete and all its p-primary subgroups have finite p-rank, for each p in ϕ, (b.2) tor(G)κ is a discrete torsion subgroup, (b.3) algebraically and topologically, G = tor(G)ϕ ⊕ L for a discrete subgroup L; moreover, L contains tor(G)κ and L/ tor(G)κ is a torsion-free discrete subgroup of finite ℤ-rank, (c) G is strongly topologically quasihamiltonian. Proof. Assume (a). Set T := tor(G) and note that by Theorem 13.12 it is open in G. Therefore T contains an open compact torsion subgroup K of G. Since the latter has finite exponent (see [54, Corollary 8.9]) one has ϕ := π(K) finite. Set κ := π(tor(G)) \ ϕ and observe the algebraic and topological direct decomposition into corresponding primary components T = Tϕ ⊕ Tκ . Since Tϕ is open, Tκ is discrete, i. e., (b.2) holds. For every p ∈ ϕ the p-primary subgroup Tp is topologically modular by (c) of Theorem 13.12. Replacing ϕ by a smaller subset of π(T) if necessary we can assume Kp to be infinite for all p ∈ ϕ. Fix any p ∈ ϕ. Then Corollary 14.10 implies all of (b.1). (b.3) Since G/T is a discrete finite ℤ-rank group one has div(G/T) ≅ ℚm for some m ≥ 0 and there is a reduced subgroup of G/T isomorphic to ℤn for some n ≥ 0. Lift a set of free generators of that group to G and, since T is open, note that they generate a discrete subgroup Z ≅ ℤn such that G = G1 ⊕ Z for an open subgroup G1 containing tor(G). From now on we may assume that G/T ≅ ℚm .

284 | 14 Strongly topologically quasihamiltonian groups Then G/Tκ has torsion subgroup T/Tκ ≅ Tϕ . From (b.1) we know that Tϕ = div(Tϕ ) ⊕ Eϕ , where Eϕ is a finite exponent open compact subgroup of Tϕ . It follows that T/Tκ ≅ Tϕ is cotorsion making it a direct summand in an algebraic and topological decomposition G1 /Tκ = T/Tκ ⊕ Q/Tκ , where the second summand is discrete and isomorphic to ℚm . Since T = Tϕ ⊕ Tκ conclude G1 = Tϕ ⊕ Q, and Q/Tκ ≅ ℚm . Summing up we have shown that L/ tor(G)κ is a torsion-free discrete group having finite ℤ-rank. Then L := Z ⊕ Q satisfies the claims in (b.3). Remark 14.33. The subgroup L in Theorem 14.32 need not be a split extension—even if G is discrete, as has been demonstrated in [54, Appendix A, Theorem A1.32] Via Pontryagin duality Theorem 14.32 implies at once the following structure theorem. Theorem 14.34. Let G be a locally compact abelian topologically modular group with nontrivial connected component G0 . Then there is a finite subset of primes ϕ of π(G/G0 ), a periodic ϕ-subgroup Tϕ of G, and a closed subgroup L of G such that all of the following statements hold: (i) Tϕ = Zϕ ⊕ Eϕ , where n

Zϕ = (⨁ ℤpp ) p∈ϕ

for nonnegative integers np and Eϕ a discrete finite exponent group, (ii) the identity component G0 is compact and connected of finite dimension and is contained in L; the factor group L/G0 is a profinite group with κ := π(L/G0 ) disjoint from ϕ. Theorem 14.35. Let G be a nonperiodic totally disconnected locally compact abelian group. The following statements are equivalent: (a) G is topologically modular, (b) G is strongly topologically quasihamiltonian. Proof. Since (b) implies (a) as has been remarked at the beginning of this chapter we only need to deduce (b) from (a).

14.5 The nonabelian situation | 285

Thus assume that G is topologically modular. Then Theorem 13.12 implies that tor(G) satisfies condition (B) of Theorem 14.22 and hence also (A), i. e., tor(G) is strongly topologically quasihamiltonian. Now let X and Y be arbitrary closed subgroups of G. Then tor(X) + tor(Y) is closed by what we just have proved. Therefore, taking Lemma 14.3 into account, there is no loss of generality to assume both X and Y to be torsion-free. Since X and Y are topologically modular their ℤ-rank is finite and we can make induction on the sum of their ℤ-ranks. Pick x ∈ X and consider the discrete and hence closed subgroup generated by 2x. Then the ℤ-rank of X/⟨2x⟩ has dropped by 1 and therefore in G/⟨2x⟩ the subgroup X/⟨2x⟩ + (Y + ⟨2x⟩)/⟨2x⟩ is closed. Hence, using Lemma 14.3 again, we deduce the closedness of X + Y. Thus G is strongly topologically quasihamiltonian. Finally we turn to groups with nontrivial connected components. Lemma 14.36. Let G be a locally compact abelian group satisfying conditions (i) and (ii) in Theorem 14.34. Then every closed subgroup X of G satisfies them. Proof. Conditions (i) and (ii) in Theorem 14.34 and the compactness of G0 imply that X/X ∩ G0 ≅ (X + G0 )/G0 ≅ Aϕ ⊕ Aκ algebraically and topologically. Since G0 has finite dimension, so has X0 and F := X ∩ G0 /X0 is a finite abelian group. Set ϕ󸀠 := ϕ \ π(F) and κ 󸀠 := κ ∪ π(F) and let L󸀠 be the preimage of Aκ in X/X0 . Then Tϕ ∩ X ⊇ Xϕ󸀠 and Xϕ󸀠 = (Tϕ󸀠 ∩ X) ⊕ L󸀠 . Thus X satisfies the conditions (i) and (ii) in Theorem 14.34. Theorem 14.37. Let G be a locally compact abelian group with nontrivial component G0 . The following conditions for G are equivalent: (a) G is topologically modular, (b) G satisfies the conditions in Theorem 14.34, (c) G is strongly topologically quasihamiltonian. Proof. Theorem 14.34 shows that (a) implies (b). Suppose (b) and let X and Y be closed subgroups. Lemma 14.36 shows that X and Y satisfy conditions (i) and (ii) of the cited theorem. Taking Lemma 14.3 into account we may in X and Y factor compact subgroups. Factoring X ∩ (L + Tϕ )Y ∩ (L + Tϕ ) allows to assume that X and Y to be discrete finite exponent subgroups of G. Then X + Y is a subgroup of the discrete subgroup Eϕ and hence is discrete. Therefore X + Y is closed and hence G is strongly topologically quasihamiltonian, i. e., (c) holds. Finally (c) implies (a) by Proposition 14.1.

286 | 14 Strongly topologically quasihamiltonian groups Let us summarize the findings of Theorems 14.8, 14.35, and 14.37. Corollary 14.38. Under the following conditions a nondiscrete locally compact abelian group G is topologically modular if and only if G is strongly topologically quasihamiltonian: (i) G is a p-group, (ii) G is totally disconnected but not periodic, (iii) G0 is not trivial. ̂ is Moreover, whenever G satisfies one of the conditions (i)–(iii), the Pontryagin dual G strongly topologically quasihamiltonian if and only if G is strongly topologically quasihamiltonian. Mukhin, in [78], has provided a complete description of nonperiodic locally compact abelian topologically modular groups. It is also said there that the subclass of locally compact abelian strongly topologically quasihamiltonian groups follows from this description. Although one easily confirms that every strongly topologically quasihamiltonian group is topologically modular we shall not make use of Mukhin’s results about topologically modular groups in the following proofs. Here we present what can be said about nonabelian strongly topologically quasihamiltonian groups and recall that every strongly topologically quasihamiltonian group with nontrivial connected components is abelian. Theorem 14.39. Let G be a locally compact nonperiodic group. The following statements are equivalent: (A) G is strongly topologically quasihamiltonian, (B) G is near abelian and the base subgroup A is a strongly topologically quasihamiltonian torsion group. Proof. Suppose first that (A) holds. Then, according to Proposition 14.1, G is topologically modular. Theorem 13.35 implies that A = comp(G) = tor(G) is torsion. As a closed subgroup of G, by Proposition 14.2, A is strongly topologically quasihamiltonian. Since we already reduced the situation to A ∩ X = A ∩ Y = {0} it follows that X and Y are discrete locally cyclic torsion-free subgroups of G. Since XA/A and YA/A are then both subgroups of the locally cyclic group G/A, it follows that XA ∩ YA ≠ A. Therefore, in particular, we have X ∩ AY ≠ {0}.

(14.2)

Observing that X ∩ AY is locally cyclic, similarly as in the proof of claim 2, one can present X = ⋃k≥1 Xk and Y = ⋃k≥1 Yk as ascending unions of respective infinite cyclic groups. Then, for k ∈ ℕ, one can find generators xk of X and yk of Y, elements ak ∈ A, and natural numbers mk such that xk = yk ak ,

m

k , xk = xk+1

m

k . and yk = yk+1

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m

k making C := ⟨ak | k ∈ ℕ⟩ a torsion locally cyclic These identities imply that ak = ak+1 subgroup, which, by Lemma 5.21, is a closed subgroup of A. By the openness of A the subgroup XA turns out to be a topological direct product, i. e.,

XA ≅ X × A, showing that XC ≅ X × C is a closed subgroup of G. In the same vein, YC turns out to be closed. Replacing X and Y by respectively XC and YC it turns out that (XC)(YC) = (XC)(CY) = XCY = (XC)Y = XY, since C is a subgroup of X. Making use of Lemma 14.3 we may factor the intersection (X ∩ C)(Y ∩ C) so that from now on we have {1} ≠ X ∩ (AY) = X ∩ Y. Factoring X ∩ Y, thereby again using the just cited lemma, allows to assume X ∩ Y = X ∩ A = Y ∩ A = {1}. Moreover, we achieved in this step that X ∩ AY = {1} as well. But then equation (14.2) implies that either X or Y is already contained in A and hence is trivial. Thus XY agrees with either X or Y and is therefore a closed subgroup of G. Hence G is strongly topologically quasihamiltonian, so that all of (A) is established. With the help of results from [78] and [79] one can provide a complete description of strongly topologically quasihamiltonian groups. Theorem 14.40. A nonperiodic locally compact group is strongly topologically quasihamiltonian if and only if it is both topologically quasihamiltonian and topologically modular. Proof. As remarked earlier, it is an easy consequence of the definitions that every strongly topologically quasihamiltonian group is topologically quasihamiltonian and topologically modular. Assume now that a group G is topologically quasihamiltonian and topologically modular. In [79, Theorem 2] it is implied that the torsion subgroup tor(G) of G is an open abelian subgroup and that G/ tor(G) is a torsion-free rank 1 group. Hence G is near abelian and has base A := tor(G). Since G is topologically modular, so is the closed subgroup tor(G), as is clear from the definitions. Therefore [78, Corollary 3] implies that tor(G) = Aκ ⊕ (Dϕ ⊕ Kϕ ) for a finite set of primes ϕ and its complement κ in π(tor(G)). Moreover, by the same corollary, Dϕ is divisible and Kϕ is compact. It follows from Theorem 14.21 that tor(G) is strongly topologically quasihamiltonian. The fact that G is strongly topologically quasihamiltonian follows now from Theorem 14.39 (A).

288 | 14 Strongly topologically quasihamiltonian groups This concludes the description of locally compact strongly topologically quasihamiltonian groups and thus marks the end of our description of some applications of the theory of near abelian locally compact groups in Part III of our treatise.

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List of symbols A∗ 210, 211 B(G) – Definition 4.29 113 χ(G) – Definition 10.24 214 CX (G) 34 div(G) 79 Div(A) 55 div(A) 55 𝔽p 138 γ(G) 210 – Definition 10.6 205 GD 162 – Sylow structure 216 GF(p) 4 GP 162 𝒢α – Definition 4.44 121 K(G) 9 Mn – Definition 8.5 182 ∇p – Example 56 ν-Theorem 43 ν(G) – Definition 2.45 39

https://doi.org/10.1515/9783110599190-016

νp (n) – Notation 3.1 47 P[m] – Definition of 141 ℙp – Definition 106 (p, q)-factor – Definition 13.16 250 ℚp – nonsplitting 89 ℛ(A) 102 – structure 110, 112 σ-(sub)group 19 σ-Sylow subgroup 19 𝕊e – Definition 4.37 117 – Definition 4.41 119 τ(G) – Definition 10.9 207 W(H; C, W ) 9 ̃ 110 ℤ ζ (G) 207 ζG – Definition 2.55 43 ℤ×p 106 ℤ(p∞ ) 106

Index abelian group – torsion 53 action – automorphic 23 approximately finite exponents – Definition 3.30 65 arithmetic progression 102 A-trivial – Definition 6.6 154 automorphic action 23 automorphism – scalar 103 base group – Definition 6.1 151 – minimal 158 – saturated 211 – structure of centralizer 152 base-discrete – Definition 9.9 195 – projective limit 196 bipartite graph – Definition 4.32 114 Braconnier – local product 41 – Theorem of 47, 49 Černikov – theorem of 32 Chabauty space – Definition 1.16 9 – of a compactly ruled group 11 – semicontinuity 11 class – good 8 commensurable 47 comp(G) 5, 48 comp0 (G) 48 compact – element 5 compactly ruled – topologically modular groups 258 – topologically quasihamiltonian p-groups 274 – Definition 1.4 6 – extensions 7 – products 7 https://doi.org/10.1515/9783110599190-017

– quotients 7 – subgroups 7 complement – Definition 1.32 16 cone 115 coreduced 75 – Definition 3.47 71 de Vries J. 30 densely divisible 84 – Definition 3.27 63 divisible – densely 63 – direct summand 79, 80 – hull 69, 71 – Main theorem 73 – minimal 72 – largest 79 – local weight 72 – p-group – structure 92 divisible subgroup – direct summand 78 edge – group 117, 119 – labeled 114 – sloping 115, 203 – vertical 115 effective part 180 element – compact 5 exponent – approximately finite 65 extension – abelian by procyclic 140 – Definition 1.7 7 – dihedral 219 – of compactly ruled groups 7 – special – Definition 4.46 123 finitely generated – Notation 2.57 43 fixed point – lifting 35

298 | Index

Frattini subgroup – definition 264 funnel 115 Gamma theorem 136 generalized quaternion group 182 Golod E. S. 6 good class 8 graph – bipartite 114 – master 114, 115 – prime 114, 121 Grigorchuk R. I. 6 Gupta N. D. 6 inductively – monothetic – classification 132 – noncompact element 133 – Notation 5.1 127 – projective limit 137 – quotient 135 – near abelian 200 – 𝒫, Definition 2.58 43 – prosolvable 44 isolated – vertex 204 – Definition 4.48 124 Iwasawa – p-factor – Definition 13.14 249 – (p, q)-factor – characterization 250 – Definition 13.16 250 Iwasawa K. 231 Khan M. S. 127 Kovács L. 30 Kulikoff – Theorem of – pro-p version 67 Kulikoff’s lemma 65 local – power – Definition 2.50 41 – dual 88 – product – ν(G) 43

– and topological M-group 258 – Automorphism of locally compact abelian group 101 – Braconnier’s theorem 49 – Definition 2.49 41 – nilfactor 43 – of topologically modular groups 253 – of Iwasawa p-factors 253 – of p-groups 41 – of topologically quasihamiltonian groups 233 – standard inductively monothetic group 130 Maier D. 222 Maschke’s theorem 36 mastergraph 115 – cone 115 – funnel 115 – prime – Definition 4.33 114 Mayer-Vietoris – formalism 15, 29 monothetic 48 – inductively 127 – standard 141 Mukhin J. N. 112, 231, 247 near abelian – A-nontrivial 154 – average trivial 185 – Definition 6.1 151 – inductive limit 196 – inductively 200 – nontrivial – structure 160 – periodic – Sylow structure 208 – p-group 159 – projective limit 194 – quotient 193 – singular 174 – special trivial 185 – structure of nonperiodic 177 – subgroup 161, 193 – Sylow structure 165, 166 – trivial – Definition 154 – structure 184, 187 Neumann P. 30

Index | 299

nilfactor – as a local product 43 – Definition 2.54 43 – splitting 39, 42 nonsingular 174 nontrivial – Definition 6.7 155 – scalar automorphism 103 Ol’shanskii A. Yu. 6 order – infinite of p-element 48 p-primary component 49 p-element – infinite order 48 periodic – Definition 1.13 8 – inductively monothetic group 132 – near abelian – structure 162 – Sylow structure 163 – scaling group 157 – solvable 9 – structure of a nontrivial near abelian group 160 – topologically quasihamiltonian – structure 234 p-group – definition 19 – divisible torsion – structure 92 – finite p-rank – structure 98 – Prüfer’s 48 – splitting 82 – topologically modular 247, 266, 267 Π-procyclic – Definition 5.9 130 – p-primary components 130 p-pure – Definition 3.37 67 p ⋅ ⋅ ⋅ q-edge 203 p-rank – and ℚp -dimension 71 – Definition 3.89 93 – Definition 3.6 51 – equal 1 128 – finite 98

primary component – Definition of 49 prime graph – associated to α 121 prime-graph – of periodic near abelian group 203 pro-Lie 195 prodiscrete 195 product – local 41 – of σ-subgroups 24 – of compactly ruled groups 7 – semidirect 13 projective limit – strong, Definition 9.6 194 Prüfer – p-group 48 Prüfer group 106 Prüfer’s example 175 pure subgroup 65 quaternion group – generalized 182 quotient – of compactly ruled group 7 rankp (G) – Definition 3.89 93 reduced 80 ring – ℛ(A) 102 – adjunction of p1 138 – of scalars – Definition 4.5 102 – units of ℤp 139 Salzmann H. 28 saturated – base group – Definition 10.19 211 SAut – Definition 4.8 103 – p-primary component 109 – Sylow structure xlii, 121 scalar – automorphism – Definition 4.8 103 – nontrivial 103 – primary decomposition 104

300 | Index

– trivial 103 scalar(s) – ring of 102 scaling subgroup – Definition 6.1 151 – existence 167, 172, 173 – non-existence 175, 222 – small 151 – small, existence 180 Schur–Zassenhaus – condition 26 – Theorem 27 semidirect product – and Open Mapping Theorem 14 – Definition 13 – of compactly ruled groups 33 Sidki S. 6 singular – Definition 7.40 174 sloping – edge 115 sloping edge – definition 203 small scaling – existence 175 small scaling subgroup 151 socle – Definition 3.4 51 special – extension – Definition 4.46 123 standard – monothetic subgroups P[m] 141 standard inductively monothetic group – Definition 5.9 130 Strong projective limit lemma 194 strongly topologically quasihamiltonian group – abelian periodic 273 – abelian torsion 272 – definition 263 – nonperiodic 286, 287 – periodic 281 – periodic semidirect product 279 𝒮𝒰ℬ(G) 9 – closedness of set of inductively monothetic groups 137 subgroup – of compactly ruled group 7 – p-pure 67

– pure 65 – standard monothetic 141 – Sylow 19 – Ulm 55 supplement – Definition 1.32 16 – existence 140, 145, 173 supplementary part 180 Sylow – base – Definition 2.60 44 – existence in a prosolvable group 44 – in compactly ruled groups 45 – inductively prosolvable group 45 – structure – of SAut 121 – of periodic near abelian group 163 – periodic near abelian 208 Sylow structure – near abelian 165, 166 – of SAut xlii, 112 Sylow subgroup 19 – central 31 – central splitting 43 – conjugacy 21, 28 – Definition 2.2 19 – normal 22, 25, 29, 31, 32 Tarski monster 6 t-cyclic 127 topologically locally finite 6 topologically modular – definition 241 – G0 ≠ {1} 284, 285 – nonperiodic 261 – periodic 258 – p-group 247, 274 topologically quasihamiltonian – definition 229 – not periodic 237 – periodic 235 – p-group 231 torsion group – abelian 53 totally disconnected – topologically modular 283 trivial – near abelian – Definition 154

Index |

– scalar automorphism 103 Ulm subgroup 55 vertex – covered 204 – isolated 204, 206 – Definition 4.48 124 – lower 115, 203

– uncovered 204, 205 – upper 115, 203 vertical – edge 115 weight 72 Weil’s Lemma 5, 111, 133, 236 Winkler R. 111

301

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