Abelian Groups: Structures and Classifications 9783110427684, 9783110432114

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Carol Jacoby and Peter Loth Abelian 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 73

Carol Jacoby and Peter Loth

Abelian Groups |

Structures and Classifications

Mathematics Subject Classification 2010 Primary: 20-2, 20Kxx, 03C52, 13C05; Secondary: 03C60, 03E10, 22B05, 22D35 Authors Dr. Carol Jacoby Jacoby Consulting Long Beach, CA USA [email protected] Prof. Dr. Peter Loth Department of Mathematics Sacred Heart University 5151 Park Avenue Fairfield, CT 06825 USA [email protected]

ISBN 978-3-11-043211-4 e-ISBN (PDF) 978-3-11-042768-4 e-ISBN (EPUB) 978-3-11-042786-8 ISSN 0179-0986 Library of Congress Control Number: 2019938386 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

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For our spouses Jerry and Alice In memoriam Rüdiger Göbel

Preface How do you get your arms around an infinite abelian group? Such groups are all around us—integers, rationals, reals, and complex numbers. Yet these examples do not even hint at the richness and complexities of abelian groups, as we will see. To help in getting through the complexity we want to focus in on some set of characteristics that completely determines the group. Specifically, we seek some invariants of a group such that if two groups share the same invariants, then they are of necessity isomorphic. As it turns out, this problem cannot be solved in general. There is a trade-off between the size of the class of groups classified and the complexity of the invariants. This means that a key part of the classification task is finding classes of groups that admit to an insightful classification. Thus, the classification problem has three parts: (1) Identify a class of groups to be classified. The broader the class the better. Much of the work in this area involves expanding the class. For example, Ulm was able to classify countable torsion groups, and later work expanded the classification to other torsion groups, and then to certain mixed groups (that is, those that are neither torsion nor torsion-free) with additional invariants. The ideal class will be closed under key operations to aid studying it. In particular, the class should be closed under direct sums and summands, and ideally G should be in the class if and only if both pα G and G/pα G are in the class. (2) Identify an invariant, usually a cardinal, that can be determined for each group. It is somewhat misleading to say “an invariant” since it may actually consist of infinitely many values. Prove that it is indeed an invariant by proving that it is unchanged under isomorphism. (3) Prove that the invariant classifies the groups in the class. Specifically, prove that if two groups in the class have the same invariants, then they are isomorphic. In some cases, we can expand the class by demanding something less than full isomorphism. This exploration leads to the discovery of interesting classes of groups, and hence to the structure problem. The goal is to describe the structure or other defining characteristics of a class of groups. For example, we will see that groups in certain classes may be written and thought of as direct sums of certain cyclic groups. Often this structure suggests invariants to use for classification. Structure and classification are often combined into uniqueness theorems that present the class as the unique one that contains certain groups, is closed under certain operations and classified by certain invariants. It can be argued that Gauss gave us the first classification result in 1801, the finite version of the primary decomposition theorem. This was even before groups and abelian groups were defined by Kronecker in 1870. The first classification result of infinite groups was due to Ulm in 1933. He developed cardinal invariants that comhttps://doi.org/10.1515/9783110427684-201

VIII | Preface pletely classified countable torsion groups. Soon after, Baer (1937) developed invariants called types that classify torsion-free groups of rank 1. This was the first work that systematically addressed groups that were not necessarily countable. Interest in abelian groups reignited in the mid-20th century with the publication of books on the topic [78, 24] by Kaplansky (1954) and Fuchs (1958), the latter containing 86 open problems. Hill extended Ulm’s classification to a larger class of torsion groups, the totally projective groups. These torsion and torsion-free results seemed to be unrelated until Warfield developed a classification theory of a class of groups that includes both, but also some groups that do not split into a direct sum of torsion and torsion-free subgroups. An apparent way to look at these mixed groups is as extensions of their torsion subgroup by a torsion-free group. Warfield flipped this around and viewed these mixed groups in terms of a torsion-free basis, characterized by the Warfield invariants, with the quotient over this basis torsion and totally projective, and thus characterized by the Ulm invariants. We extend this work in two ways. One is to weaken the requirement for isomorphism. Barwise and Eklof extended Ulm’s theorem to all torsion groups up to partial isomorphism. Since partial isomorphism implies isomorphism on countable groups, this is a true generalization. In Chapter 5, we define a class of groups that includes the Warfield groups and classify it up to partial isomorphism. Partial isomorphism has an interesting model-theoretic counterpart that we explore. The other extension looks at topological groups. In Chapters 6 and 7, we use Pontrjagin duality to determine the dual counterparts of various constructs in topological groups. Every group result has a corresponding result in compact abelian groups. In particular, this gives us classifications for various classes of compact abelian groups. The classes of groups all have a range of characterizations. We explore each class in terms of presentations, homology, increasing sequences of subgroups, decomposition bases, and many others. Many of the chapters culminate in a list of varied characterizations that are proved equivalent. We assume the reader is familiar with the definition of group, and key constructs, such as subgroups, factor groups, homomorphisms, and isomorphisms. When we use the word “group”, we will mean additive abelian group. At times, we will consider groups as modules over the integers or over ℤp , a ring that will be useful for looking at groups one prime at a time. Necessary background in the key characteristics of infinite abelian groups is provided in Chapter 1.

Acknowledgments We would like to express our gratitude to the memory of the late Professor Rüdiger Göbel of the University of Duisburg–Essen. He greatly impacted the direction of Peter Loth’s research and initiated a research project that led to our subsequent collaboration. Carol Jacoby would like to thank him for his generous help and encouragement in incorporating and extending her previous research. She gratefully acknowledges Professor Emeritus Paul Eklof of the University of California, Irvine, for recognizing the synergy possible between this research and her previous work, and providing suggestions and connections to make it happen. This project was partially supported by a University Research/Creativity Grant (URCG) from Sacred Heart University. We wish to thank Walter de Gruyter Verlag for the publication of this book.

https://doi.org/10.1515/9783110427684-202

Contents Preface | VII Acknowledgments | IX 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17

Basic characteristics of abelian groups | 1 Preliminaries | 1 Commutative diagrams | 5 The torsion part of a group | 8 Independence and rank | 9 Free abelian groups | 11 Divisible groups | 13 Purity, p-independence and p-basic subgroups | 17 Bounded groups | 21 Finitely generated groups | 22 Heights and isotypeness | 23 Characteristics and examples of heights | 27 Proper elements and height-preserving homomorphisms | 29 Ulm invariants | 31 Ulm’s theorem | 35 Completely decomposable torsion-free groups | 37 Hom and Ext | 40 Tensor products and p-localizations | 53

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11

Simply presented groups | 61 p-Nice submodules | 61 Families of subgroups and the third axiom of countability | 63 Classification of p-groups with nice systems | 68 Totally projective p-groups and generalized Prüfer groups | 72 Defining relations and simple presentations | 75 Simply presented torsion-free groups | 78 Richman–Walker groups and variations | 79 Simply presented torsion groups | 82 Balanced-projective p-groups | 94 The class of simply presented p-groups | 98 Going further | 100

3 3.1 3.2

Warfield groups | 103 Groups with decomposition bases | 103 The local classification theorem | 107

XII | Contents 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17

The class of Warfield modules | 114 The p-localization of a group | 116 Compatible Ulm matrices | 120 ∗ The subgroups G(α , p), G(M∗ ), G(M∗ , p) and ∗-valuated coproducts | 121 The global classification theorem | 125 Nice subgroups | 132 Quasi-sequentially nice subgroups | 138 Primitive elements and ∗-decomposition sets | 143 k-Subgroups and k-basic subgroups | 153 Knice subgroups and k-groups | 162 k-Subgroups and strong k-subgroups | 169 Sequentially-pure-projective groups | 172 Locally compatible and separable subgroups | 176 The class of global Warfield groups | 180 Going further | 181

4 4.1 4.2 4.3 4.4 4.5 4.6

Infinitary logic looks at groups | 183 Partial isomorphisms | 183 The language L∞ω and Lλ∞ω | 185 Karp’s theorem | 189 Expressibility in L∞ω | 191 The classification of torsion groups up to partial isomorphism | 195 Going further | 200

5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10

Groups with partial decomposition bases | 203 Partial decomposition bases | 203 Classification of local groups up to partial isomorphism | 211 Classification of general groups up to partial isomorphism | 216 Classification up to an ordinal | 220 PDB groups and k-groups | 235 Partial subbases and properties of ∗-decomposition sets | 238 PDB groups and Warfield groups | 244 The class of PDB groups | 250 Infinitary logic perspective | 253 Going further | 260

6 6.1 6.2 6.3 6.4

Characters and Pontrjagin duality of locally compact abelian groups | 263 Preliminaries | 263 Character groups and Pontrjagin duality | 267 Discrete and compact groups | 269 Annihilators | 270

Contents | XIII

6.5 6.6 6.7 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8

Proper homomorphisms and extensions of LCA groups | 276 Topological p-groups | 278 Going further | 282 Classifications of compact abelian groups | 285 Dimension, p-dimension and p-order | 285 Compact torsion and torsion-free groups | 287 Simply given groups | 288 Simply given totally disconnected groups | 289 Simply given connected groups | 295 Direct factors of simply given groups | 296 Groups with partial quasi-decompositions | 305 Going further | 309

Symbols | 311 Bibliography | 315 Index | 321

1 Basic characteristics of abelian groups This introductory chapter presents the background and key characteristics needed to classify abelian groups. Key distinguishing characteristics of groups and their subgroups include reduced, divisible, torsion, and torsion-free. The key characteristic of a group element is its height, and we also often focus on elements in the p-socle for various primes p. Key techniques that will be used throughout the development are introduced here: localization, breaking the group into direct summands, extension theorems, commutative diagrams, short exact sequences, and basic homology. We introduce the two basic classical classification results, Ulm’s theorem for countable torsion groups and Baer’s theorem for completely decomposable torsion-free groups. This chapter may be used alone as a short introduction to the study of abelian groups.

1.1 Preliminaries In this section, we introduce some basic notations, definitions, and facts, mostly without proof. For background and more details, we refer the reader to the book by Fuchs, “Abelian Groups” [29]. Throughout, we accept the ZFC axioms of set theory (Zermelo– Fraenkel with the Axiom of Choice). Let X be a partially ordered set, that is, a set with a binary relation ≤ such that for all x, y, z ∈ X, we have: 1. x ≤ x; 2. x ≤ y and y ≤ x imply x = y; 3. x ≤ y and y ≤ z imply x ≤ z. We write x < y whenever x ≤ y and x ≠ y. An element m ∈ X is called minimal if x ≤ m with x ∈ X implies m = x. Likewise, an element m󸀠 ∈ X is said to be maximal if m󸀠 ≤ x with x ∈ X implies m󸀠 = x. The set X satisfies the minimum (or maximum) condition if every nonempty subset of X contains a minimal (or maximal) element. An element a ∈ X is called the least element (or smallest element) of X if a ≤ x for all x ∈ X, and b ∈ X is said to be the greatest element (or largest element) of X if x ≤ b for all x ∈ X. A subset C of X is called a totally ordered set or a chain if x ≤ y or y ≤ x whenever x, y ∈ C. An element u ∈ X is called an upper bound of C in X if x ≤ u for all x ∈ C. We will make frequent use of the following lemma, which is equivalent to the Axiom of Choice. Zorn’s lemma. Let X be a nonempty partially ordered set, in which every chain has an upper bound in X. Then X contains at least one maximal element. A partially ordered set is called well-ordered if every nonempty subset has a least element. Note that a well-ordered set is automatically totally ordered. If C is a https://doi.org/10.1515/9783110427684-001

2 | 1 Basic characteristics of abelian groups nonempty subset of a well-ordered set X such that C has an upper bound in X, then C possesses a least upper bound, called the supremum of C. It is denoted by sup C. Many of the invariants that are used to classify groups are based on ordinals. A set S is called transitive if for all x ∈ S and y ∈ x we have y ∈ S. Then ordinals are defined to be transitive sets whose elements are transitive sets. Since elements of an ordinal α are ordinals, it will be convenient to write β < α instead of β ∈ α, and we write α = {β : β < α}. Notice that α is well-ordered by ≤, thus every set of ordinals is well-ordered. In particular, any decreasing sequence of ordinals becomes stationary. Finite ordinals are denoted by 0, 1, 2, 3, . . . . Specifically, we define 0 = 0 and n + 1, the successor of the finite ordinal n, is defined as {n, {n}}. Then ω = {0, 1, 2, 3, . . . } is the smallest infinite ordinal, and n < ω or n ∈ ω simply means that n is a nonnegative integer. The set of all positive integers (also called natural numbers) is denoted by ℕ. We assume that the reader is familiar with ordinal arithmetic. Recall the following properties of ordinals. If α, β, and γ are ordinals, then 1. α + (β + γ) = (α + β) + γ 2. α ≤ β if and only if there is an ordinal δ such that α + δ = β 3. β ≤ α + β 4. Left cancelation: If α + β ≤ α + γ, then β ≤ γ 5. Division algorithm: If β ≠ 0, then there are unique ordinals ψ and ν such that α = βψ + ν and ν < β. In particular, every ordinal can be written as α = ωδ + n with an ordinal δ and n < ω. Note that ordinal arithmetic is not commutative. For example, 1 + ω = ω, whereas ω + 1 > ω. Note also that right cancelation is invalid. For example, 1 + ω = 2 + ω. Let Ord denote the class of all ordinals, and let Ord∞ = Ord ∪{∞}. An ordinal of the form α = σ + 1 for some ordinal σ is called a successor ordinal; in this case, we say that α − 1 exists. An ordinal > 0 is called a limit ordinal if it is not a successor ordinal. A subset A of a limit ordinal α is called cofinal in α if sup A = α. We adopt the convention that α < ∞ for all α ∈ Ord∞ . For any α, β ∈ Ord∞ , we write α ≨ β whenever α < β and α ≠ β, i. e., if ∞ ≠ α < β. Throughout this book we will use the notation α1 ∧ ⋅ ⋅ ⋅ ∧ αn

or

min{α1 , . . . , αn }

to denote the minimum of the ordinals α1 , . . . , αn . The choice will depend on which is clearer in the context. In particular, we will usually use the first notation whenever only two ordinals are involved. Collections of sets are frequently indexed by ordinals; for instance, by {Sα }α 0, S(α) is true whenever S(β) is true for all β < α. Then S(α) is true for all ordinals α. Proof. Assume that S(γ) is false for some ordinal γ. Then the set X of all ordinals β ≤ γ, for which S(β) is false is nonempty; so X has a least element α. Since α > 0 by (1) and S(β) is true for all β < α, it follows from (2) that S(α) is true, a contradiction. We will be interested in cardinality to compare the sizes of infinite groups. Two sets S and T are called equinumerous, written |S| = |T|, if there is a bijection from S onto T. Then cardinals can be viewed as ordinals α such that |β| ≠ |α| for all β < α. If S is a set and κ is a cardinal with |S| = |κ|, we simply write |S| = κ and call κ the cardinality of S. The first infinite cardinal is ℵ0 = ω. Note that for cardinals κ ≤ λ, we have κ + λ = λ = κλ whenever λ is infinite. In this book, all groups are assumed to be abelian, so the word “abelian” is mostly omitted. A subgroup A of a group G is called a proper subgroup if A ≠ G. By ℤ, we mean the group or ring of integers; ℙ is the set of primes in ℤ, and ℚ is the group or ring of rational numbers. For a prime p, ℤp denotes the group or ring of integers localized at p, that is, ℤp = {

m : m, n ∈ ℤ and p ∤ n}. n

By R, we will always mean an arbitrary principal ideal domain with 1, unless stated otherwise. Recall that any elements a, b ∈ R have a greatest common divisor (gcd) and that there exist k, l ∈ R such that ka + lb = gcd(a, b). An R-module is an abelian group G, together with a map R × G → G, written (r, x) 󳨃→ rx, such that (r + s)x = rx + sx r(x + y) = rx + ry

(rs)x = r(sx) 1x = x

for all r, s ∈ R and x, y ∈ G. If G and H are R-modules, then an R-module homomorphism of G to H is a map α : G → H, satisfying α(x + y) = α(x) + α(y)

and

α(rx) = r(α(x))

for all x, y ∈ G and r ∈ R. We often refer to α simply as a homomorphism.

4 | 1 Basic characteristics of abelian groups We will find it useful to consider groups as special cases of R-modules. In particular, groups are ℤ-modules. We will also find it useful to focus on a particular prime p by considering ℤp -modules. Then a ℤp -module is also called a p-local group. Notice that a p-group can be regarded as a p-local group. A group homomorphism G → H is automatically a ℤp -module homomorphism if G and H are p-local groups. Let G and H be R-modules. For a homomorphism α : G → H, the image (also called range) of α is denoted by im α or α(G). The kernel of α is ker α = {x ∈ G : α(x) = 0}, and by the cokernel of α, we mean coker α = H/imα. If α is injective, then we sometimes call α a monomorphism. A surjective homomorphism is called an epimorphism. If α : G → H is a bijective homomorphism, it is said to be an isomorphism, and we write G ≅ H. A homomorphism ν : G → G is called an endomorphism. For example, the zero map 0G : x 󳨃→ 0, and the identity map 1G : x 󳨃→ x are endomorphisms. In the case where the endomorphism ν is an isomorphism, it is called an automorphism. The composite of two homomorphisms α : G → H and β : H → K is written βα : G → K. For a prime p ∈ R and a nonnegative integer n, we define pn G = {pn x : x ∈ G}

and G[pn ] = {x ∈ G : pn x = 0}.

Then G[p] = {x ∈ G : px = 0} is called the p-socle of G, and G is called p-bounded if G = G[p]. Notice that G[p] can be regarded as a vector space over the field R/(p). For a subset S of G, the submodule of G generated by S is written ⟨S⟩. Specifically, ⟨S⟩ consists of all linear combinations of elements of S. We will use this same notation for submodules generated by combinations of subsets and elements, such as ⟨S, x⟩ or ⟨x1 , . . . , xn ⟩. Of particular interest is the cyclic submodule of G generated by an element x ∈ G, written ⟨x⟩. The trivial R-module is written {0} or 0. If A and B are submodules of G, the submodule C of G generated by A and B is written C = A + B. If in addition A ∩ B = 0, we write C = A ⊕ B and call C the (internal) direct sum of its submodules A and B. Then A is said to be a direct summand of C. Likewise, the submodule of G generated by a family of submodules Ai (i ∈ I) is denoted by A = ∑i∈I Ai . If the submodules Ai satisfy Ai ∩ ∑j=i̸ Aj = 0 for all i ∈ I, then we call A the (internal) direct sum of submodules Ai and write A = ⨁i∈I Ai . If I is finite, say I = {1, . . . , n}, then we often write A = A1 ⊕ ⋅ ⋅ ⋅ ⊕ An . In case Ai are cyclic submodules generated by elements xi (i ∈ I), we sometimes use the notation A = ∑x∈X ⟨x⟩ (A = ⨁x∈X ⟨x⟩ if the sum is direct), where X = {xi : i ∈ I}. Now suppose that Ai are any given R-modules (i ∈ I). Then the direct product ∏i∈I Ai becomes an R-module via (xi )i∈I + (yi )i∈I = (xi + yi )i∈I and r(xi )i∈I = (rxi )i∈I for r ∈ R. Its submodule A = {(xi )i∈I : xi = 0 for almost all i ∈ I} is called the (external) direct sum of modules Ai . Notice that there is a natural isomorphism between internal and external direct sums. Thus, we simply write A = ⨁i∈I Ai as before, and again, we may write A = A1 ⊕ ⋅ ⋅ ⋅ ⊕ An whenever I = {1, . . . , n}.

1.2 Commutative diagrams | 5

We make frequent use of the easily verified modular law: (A + B) ∩ C = A + (B ∩ C) whenever A, B and C are submodules of G with A ⊆ C. For a submodule A of G, the factor group G/A becomes an R-module by defining r(x + A) = rx + A for r ∈ R. Recall Noether’s isomorphism theorems for R-modules: 1. If α : G → H is an epimorphism, then G/ ker α ≅ H. 2. If A and B are submodules of G, then (A + B)/B ≅ A/(A ∩ B). 3. If A and B are submodules of G with A ⊆ B, then G/B ≅ (G/A)/(B/A). A sequence of R-modules and homomorphisms αn

α2

α1

A1 󳨀→ A2 󳨀→ ⋅ ⋅ ⋅ → An 󳨀→ An+1

(n ≥ 2)

is called exact at Ai (with i ∈ {2, . . . , n}) if im αi−1 = ker αi . The sequence is called exact if it is exact at Ai for all i = 2, . . . , n. By a short exact sequence, we mean an exact sequence of the form β

α

0 → A 󳨀→ B 󳨀→ C → 0. In this case, we have A ≅ α(A) and C ≅ B/α(A). Exercises. 1. Let X be a partially ordered set, and let a ∈ X. If a is a least member of X, then a is minimal. The converse holds whenever X is totally ordered. 2. Show that ℤp /pn ℤp ≅ ℤ/pn ℤ. 3. Show that | ⨁i∈I Gi | = ∑i∈I |Gi | whenever |I| ≥ ℵ0 . α

4. (a) 0 → A 󳨀→ B is exact if and only if α is a monomorphism. β

5.

(b) B 󳨀→ C → 0 is exact if and only if β is an epimorphism. α (c) 0 → A 󳨀→ B → 0 is exact if and only if α is an isomorphism. If 0 → Ai → Bi → Ci → 0 are exact sequences (i ∈ I), then the induced sequences 0 → ⨁i∈I Ai → ⨁i∈I Bi → ⨁i∈I Ci → 0 and 0 → ∏i∈I Ai → ∏i∈I Bi → ∏i∈I Ci → 0 are exact as well.

1.2 Commutative diagrams A diagram of R-modules and homomorphisms is called commutative, and we say the diagram commutes if different paths along directed arrows from one module to another module result in the same composite homomorphisms. For example, the diagram α1

0

→

A1 ↓ ϕ1

󳨀→

0

→

B1

󳨀→

β1

α2

A2 ↓ ϕ2

󳨀→

B2

󳨀→

β2

A3 ↓ ϕ3

→

0

B3

→

0

6 | 1 Basic characteristics of abelian groups is commutative exactly if the left square commutes (that is, if β1 ϕ1 = ϕ2 α1 ) and the second square commutes (that is, if β2 ϕ2 = ϕ3 α2 ). β

α

Lemma 1.2.1. Let 0 → A 󳨀→ B 󳨀→ C be an exact sequence and ϕ : G → B a homomorphism. Then there is a homomorphism ν : G → A such that the diagram ν

0

A

→

↙

G ↓ϕ

󳨀→

B

α

β

C

󳨀→

is commutative if and only if βϕ = 0. Proof. If such a map ν exists, then βϕ = βαν = 0. To prove the converse, assume that βϕ = 0, and define a map ν : G → A by g 󳨃→ a if α(a) = ϕ(g). Such an a exists since βϕ(g) = 0 implies ϕ(g) ∈ ker β = im α. This map is well-defined since α is injective. Clearly, ν is a homomorphism and αν = ϕ. α

β

Lemma 1.2.2. If A 󳨀→ B 󳨀→ C → 0 is exact and ϕ : B → G is a homomorphism, then there is a homomorphism ν : C → G such that the diagram A

α

󳨀→

B ϕ↓ G

β

󳨀→ ↙ν

C

→

0

is commutative if and only if ϕα = 0. Proof. If there is such a map ν, then ϕα = νβα = 0. Conversely, suppose that ϕα = 0, and define ν : C → G by c 󳨃→ ϕ(b) if β(b) = c with b ∈ B, which can always be done since im β = C. To verify that ν is well-defined, assume that β(b󸀠 ) = c with b󸀠 ∈ B. Then b − b󸀠 ∈ ker β = im α ⊆ ker ϕ since ϕα = 0, thus ϕ(b) = ϕ(b󸀠 ). The map ν is a homomorphism, and we have νβ = ϕ, as required. The next result is known as the Five lemma: Lemma 1.2.3. If the diagram α1

A1 ↓ ϕ1

󳨀→

B1

󳨀→

β1

α2

A2 ↓ ϕ2

󳨀→

B2

󳨀→

β2

α3

A3 ↓ ϕ3

󳨀→

B3

󳨀→

β3

α4

A4 ↓ ϕ4

󳨀→

B4

󳨀→

β4

A5 ↓ ϕ5 B5

with exact rows is commutative, then we have: 1. Suppose ϕ1 is an epimorphism and ϕ5 is a monomorphism. If ϕ2 and ϕ4 are isomorphisms, then so is ϕ3 . 2. Suppose that ϕ1 is an epimorphism. If ϕ2 and ϕ4 are monomorphisms, then so is ϕ3 . 3. Suppose that ϕ5 is a monomorphism. If ϕ2 and ϕ4 are epimorphisms, then so is ϕ3 .

1.2 Commutative diagrams | 7

Proof. (1) follows from (2) and (3). Now assume that ϕ1 , ϕ2 , and ϕ4 are maps as in (2). To prove that ϕ3 is injective, let a ∈ ker ϕ3 . Then ϕ4 α3 (a) = β3 ϕ3 (a) = 0, thus injectivity of ϕ4 yields α3 (a) = 0. Since im α2 = ker α3 , we have a = α2 (a2 ) for some a2 ∈ A2 ; hence, β2 ϕ2 (a2 ) = ϕ3 α2 (a2 ) = ϕ3 (a) = 0. Then im β1 = ker β2 shows that ϕ2 (a2 ) = β1 (b1 ) for some b1 ∈ B1 . Now ϕ1 is surjective, thus there is an a1 ∈ A1 such that b1 = ϕ1 (a1 ). Then ϕ2 α1 (a1 ) = β1 ϕ1 (a1 ) = β1 (b1 ) = ϕ2 (a2 ), and we obtain α1 (a1 ) = a2 since ϕ2 is injective. But then a = α2 (a2 ) = α2 α1 (a1 ) = 0. Therefore, ϕ3 is injective. Next we assume that ϕ2 , ϕ4 , and ϕ5 are maps as in (3). We let b ∈ B3 and show that b ∈ imϕ3 . Since ϕ4 is surjective, there is an a4 ∈ A4 such that β3 (b) = ϕ4 (a4 ), thus ϕ5 α4 (a4 ) = β4 ϕ4 (a4 ) = β4 β3 (b) = 0. It follows that α4 (a4 ) = 0 since ϕ5 is injective. Since ker α4 = im α3 , there is an a3 ∈ A3 such that α3 (a3 ) = a4 ; hence, β3 ϕ3 (a3 ) = ϕ4 α3 (a3 ) = ϕ4 (a4 ) = β3 (b). Then ker β3 = im β2 shows that there is a b2 ∈ B2 , satisfying ϕ3 (a3 ) − b = β2 (b2 ). By assumption, ϕ2 is surjective; so b2 = ϕ2 (a2 ) for some a2 ∈ A2 . Thus, ϕ3 α2 (a2 ) = β2 ϕ2 (a2 ) = β2 (b2 ) = ϕ3 (a3 ) − b, and we obtain b ∈ imϕ3 , as desired. The 3 × 3-Lemma will be needed: Lemma 1.2.4. Suppose that the following diagram is commutative and all columns are exact: 0 ↓

α1

0

→

A1 ↓ ϕ1

󳨀→

0

→

A2 ↓ ϕ2

󳨀→

0

→

A3 ↓ 0

α2

α3

󳨀→

0 ↓

β1

B1 ↓ ψ1

󳨀→

B2 ↓ ψ2

󳨀→

B3 ↓ 0

β2

β3

󳨀→

0 ↓ C1 ↓ χ1

→

0

C2 ↓ χ2

→

0

→

0

C3 ↓ 0

If the first two or the last two rows are exact, then all three rows are exact. Proof. Assuming exactness of the first two rows, we show that the third row is exact. To prove that α3 is injective, let a ∈ ker α3 . Then ϕ2 (a2 ) = a for some a2 ∈ A2 ; hence, ψ2 α2 (a2 ) = α3 ϕ2 (a2 ) = 0. Since ker ψ2 = im ψ1 , there is a b1 ∈ B1 such that α2 (a2 ) = ψ1 (b1 ), thus χ1 β1 (b1 ) = β2 ψ1 (b1 ) = β2 α2 (a2 ) = 0. By injectivity of χ1 , we obtain β1 (b1 ) = 0. Then im α1 = ker β1 shows that b1 = α1 (a1 ) for some a1 ∈ A1 , which implies that α2 (a2 ) = ψ1 (b1 ) = ψ1 α1 (a1 ) = α2 ϕ1 (a1 ). But then a2 − ϕ1 (a1 ) ∈ ker α2 , thus a2 = ϕ1 (a1 ). It follows that a = ϕ2 (a2 ) = ϕ2 ϕ1 (a1 ) = 0; hence, α3 is injective. Next we show that im α3 = ker β3 . Letting a ∈ A3 , we have ϕ2 (a2 ) = a for some a2 ∈ A2 . Then β3 α3 (a) = β3 α3 ϕ2 (a2 ) = β3 ψ2 α2 (a2 ) = χ2 β2 α2 (a2 ) = 0 shows that im α3 ⊆ ker β3 . Now let b ∈ ker β3 . Then b = ψ2 (b2 ) for some b2 ∈ B2 , hence, χ2 β2 (b2 ) =

8 | 1 Basic characteristics of abelian groups β3 ψ2 (b2 ) = 0. Since ker χ2 = im χ1 , we have β2 (b2 ) = χ1 (c1 ); for some c1 ∈ C1 . Then there is an element b1 ∈ B1 such that β1 (b1 ) = c1 , and we obtain β2 (b2 ) − β2 ψ1 (b1 ) = β2 (b2 ) − χ1 β1 (b1 ) = β2 (b2 ) − β2 (b2 ) = 0, thus b2 − ψ1 (b1 ) ∈ ker β2 = im α2 . Consequently, we have b2 − ψ1 (b1 ) = α2 (a2 ) for some a2 ∈ A2 , and it follows that α3 ϕ2 (a2 ) = ψ2 α2 (a2 ) = ψ2 (b2 − ψ1 (b1 )) = ψ2 (b2 ) = b. Thus, b ∈ imα3 . To prove surjectivity of β3 , let c ∈ C3 . Then there is a b2 ∈ B2 such that c = χ2 β2 (b2 ) = β3 ψ2 (b2 ), therefore c ∈ imβ3 . The proof of the second assertion is very similar. Exercises. 1. Show that the map ν : G → A in Lemma 1.2.1 is uniquely determined. 2. Show that the map ν : C → G in Lemma 1.2.2 is uniquely determined.

3.

ϕ

A homomorphism ϕ : A1 → A2 induces an exact sequence 0 → ker ϕ → A1 󳨀→ A2 → coker ϕ → 0. 4. Prove the Snake lemma: Let

0

→

A1 ↓ϕ A2

→ →

B1 ↓ψ B2

→ →

C1 ↓χ C2

→

0

be a commutative diagram with exact rows. Then there is an exact sequence ker ϕ → ker ψ → ker χ → coker ϕ → coker ψ → coker χ.

1.3 The torsion part of a group Let G be a group. By the torsion part of G, we mean the set tG = {x ∈ G : nx = 0 for some nonzero integer n}. If G = tG, we call G torsion, and if tG = 0, then we say that G is torsion-free. If A is a subgroup of G such that G/A is torsion, we sometimes say that G is torsion over A. The group G is called mixed if G is neither torsion nor torsion-free; that is, if G has both nonzero elements of finite order and elements of infinite order. Theorem 1.3.1. Let G be a group. Then its torsion part tG is a subgroup of G, and G/tG is torsion-free. Proof. Clearly, tG is nonempty since it contains 0. If x, y ∈ tG, then rx = 0 = sy for some nonzero integers r, s, which yields rs(x − y) = 0, thus x − y ∈ tG. Therefore, tG is a subgroup of G. To show that G/tG is torsion-free, let g ∈ G and assume that ng ∈ tG for some nonzero integer n. Then mng = 0 for some nonzero integer m, thus g ∈ tG. It follows that G/tG is torsion-free.

1.4 Independence and rank | 9

For a given prime p, the p-torsion part of a group G is defined to be the set tp G = {x ∈ G : pn x = 0 for some nonnegative integer n}. In the case tp G = 0, we say that G has no p-torsion. Theorem 1.3.2. Let G be a group. Then tp G is a subgroup of G for all primes p, and we have tG = ⨁p∈ℙ tp G. Proof. Each set tp G is nonempty since it contains 0. If x, y ∈ tp G, then pr x = 0 = ps y for some r, s < ω; hence, pr+s (x − y) = 0, and therefore, x − y ∈ tp G. Thus, tp G is a subgroup of G. n n Let g ∈ tG. Then ng = 0 for some positive integer n, and we write n = p1 1 ⋅ ⋅ ⋅ pk k , 󸀠 where p1 , . . . , pk are distinct primes and n1 , . . . , nk < ω. For any i ∈ {1, . . . , k}, let ni = n n/pi i . Then n󸀠i g ∈ tpi G. Since n󸀠1 , . . . , n󸀠k are relatively prime, there are integers s1 , . . . , sk such that s1 n󸀠1 + ⋅ ⋅ ⋅ + sk n󸀠k = 1, thus k

g = s1 n󸀠1 g + ⋅ ⋅ ⋅ + sk n󸀠k g ∈ ∑ tpi G, i=1

and we obtain tG = ∑p∈ℙ tp G. If p is a prime and x ∈ tp G ∩ ∑q=p̸ tq G, then pk x = 0 = mx for some k < ω and integer m with p ∤ m, which shows that x = 0. Therefore, we have tG = ⨁p∈ℙ tp G. By the theorem above, the study of torsion groups is reduced to that of p-groups. Exercises. 1. A group G is called elementary if the order of each element of G is a square-free integer. Show: (a) A group is elementary if and only if it is a direct sum of cyclic groups of prime orders. (b) Every subgroup of an elementary group is a direct summand. 2. A group is said to split if its torsion part is a direct summand. Show the following: If P is an infinite set of primes, then ∏p∈P ℤ/pℤ does not split.

1.4 Independence and rank A set X of nonzero elements of a group G is called independent if for all x1 , . . . , xn ∈ X and integers a1 , . . . , an , a1 x1 + ⋅ ⋅ ⋅ + an xn = 0 implies that a1 x1 = ⋅ ⋅ ⋅ = an xn = 0.

10 | 1 Basic characteristics of abelian groups Note that—in this case—we have ai = 0 provided that the order of xi is infinite. The following fact is immediate: Lemma 1.4.1. Let G be a group and X a set of independent elements of G. Then X is independent if and only if the sum ∑x∈X ⟨x⟩ is direct. We let IG denote the set of all elements of G of infinite order. By Zorn’s lemma, any independent subset of IG is contained in some maximal independent subset of IG . Likewise, for any given prime p, any independent subset of the p-torsion part tp G is contained in a maximal independent subset of tp G. We have the following: Theorem 1.4.2. Let G be a group. Then any two maximal independent subsets of IG have the same cardinality. Likewise, for a given prime p, any two maximal independent subsets of tp G have the same cardinality. Proof. Notice that a set {x1 , . . . , xk } ⊆ IG is independent in G exactly if {x1 +tG, . . . , xk +tG} is independent in the torsion-free group G/tG. Therefore, for the first claim, we may assume that G is torsion-free. Let X = {xi : i ∈ I} be a maximal independent set in G. Letting g be a nonzero element of G, we have rg = a1 xi1 +⋅ ⋅ ⋅+at xit for some r ∈ ℕ and 0 ≠ ak ∈ ℤ, ik ∈ I (k = 1, . . . , t); so we can associate with g the tuple (r, a1 , . . . , at , i1 , . . . , it ). Note that if 0 ≠ g 󸀠 ∈ G is associated with the same tuple, then r(g − g 󸀠 ) = 0 implies that g = g 󸀠 since G is torsion-free. Thus, |G| ≤ |X| ⋅ ℵ0 ; so since |X| ≤ |G|, we obtain |X| = |G| whenever X is infinite. If X is finite, we write X = {x1 , . . . , xn }. Then if Z = {z1 , . . . , zm } is any independent set in G, we have ki zi ∈ ⟨X⟩ for some ki ∈ ℕ (i = 1, . . . , m) by the maximality of X. Assume that m > n and write ki zi = ∑nj=1 λij xj (i = 1, . . . , m). Then there is a nonzero (μ1 , . . . , μm ) ∈ ℤm such that 0 λm1 λ11 . . . . μ1 ( . ) + ⋅ ⋅ ⋅ + μm ( . ) = ( ... ) , λmn 0 λ1n and we have ∑i μi ki zi = ∑i,j μi λij xj = ∑j (∑i μi λij )xj = 0, contradicting independence of {z1 , . . . , zm }. Thus, |Z| ≤ |X|. Now let Y be any maximal independent subset of G. In the case where X is finite, we obtain |Y| ≤ |X|, and by symmetry, we have |X| ≤ |Y|; hence, |X| = |Y|. In the case where X is infinite, Y cannot be finite by the same argument we just used, thus |X| = |G| = |Y|. It remains to show that any two maximal independent subsets of tp G have the same cardinality, so we can assume that G is a p-group. Then a subset {x1 , . . . , xk } of G is independent if and only if {pn1 −1 x1 , . . . , pnk −1 xk } is independent, where pni is the order of xi (i = 1, . . . , k); thus, it suffices to consider maximal independent subsets X of the p-socle G[p]. Now G[p] can be considered a ℤ/pℤ-vector space, so X is independent exactly if it is linearly independent in the vector space G[p]. It follows that the

1.5 Free abelian groups |

11

cardinality of any maximal independent subset of G coincides with the dimension of G[p]. The cardinality of a maximal independent subset of IG is called the torsion-free rank of G, written r0 (G). Likewise, if we restrict ourselves to subsets of the p-torsion part tp G for any given prime p, the cardinality rp (G) of a maximal independent subset of tp G is called the p-rank of G. We define rp,0 (G) = rp (G)+r0 (G). The rank of G, written r(G), is defined as the sum of its torsion-free and p-ranks for all primes p. Note that the rank of a torsion-free group is the torsion-free rank, and the rank of a p-group is the p-rank. By Theorem 1.4.2 and its proof, we have the following: Corollary 1.4.3. Let G be a group and p a prime. Then the cardinals r0 (G), rp (G), rp,0 (G), and r(G) are invariants of G. We have r0 (G) = r0 (G/tG) and rp (G) = rp (G[p]). The torsion-free rank of a group can be described as follows: Theorem 1.4.4. Let G be a group and m a cardinal. Then r0 (G) = m if and only if G contains a subgroup A ≅ ⨁m ℤ such that G/A is torsion. Proof. If r0 (G) = m, then G has a maximal independent subset X of IG with |X| = m. By Lemma 1.4.1, A = ⟨X⟩ is a direct sum of infinite cyclic groups. Assume that g ∈ G \ X is an element of infinite order. Then ⟨g⟩ ∩ ⟨X⟩ ≠ 0 by the maximality of X, thus ng ∈ ⟨X⟩ for some positive integer n. Therefore, G/A is torsion. Conversely, suppose that G/A is torsion for some subgroup A ≅ ⨁m ℤ of G. Then we can write A = ⨁x∈X ⟨x⟩ for a subset X of A such that each element of X is nonzero and has infinite order. Since G/⟨X⟩ is torsion, it follows that X is a maximal independent set in IG . Exercises. 1. Let A be a subgroup of G. Then: (a) r(G) ≤ r(A) + r(G/A). (b) r0 (G) = r0 (A) + r0 (G/A). 2. If G = ∑i∈I Ai , then r(G) ≤ ∑i∈I r(Ai ). Equality holds whenever the sum is direct. 3. A subgroup E of a group G is said to be essential if E ∩ A ≠ 0 for each subgroup A ≠ 0 of G. Show that an independent set X of elements of G is maximal if and only if ⟨X⟩ is an essential subgroup of G.

1.5 Free abelian groups A group is called free if it is a direct sum of infinite cyclic groups. Thus, a group F is free exactly if it has a subset X such that F = ⨁⟨x⟩ x∈X

12 | 1 Basic characteristics of abelian groups with ⟨x⟩ ≅ ℤ for all x ∈ X. It is clear that the elements of F can be written uniquely as expressions z = a1 x1 + ⋅ ⋅ ⋅ + an xn

(∗)

with coefficients a1 , . . . , an ∈ ℤ\{0} and distinct x1 , . . . , xn ∈ X, and that addition of two such expressions is carried out by adding coefficients belonging to the same elements of X. Alternatively, given any set X, the group F could have been defined to be the set of all formal expressions (∗), together with the aforementioned addition. We call F the free group on the set X, and say that X is a set of free generators. The following basic fact will be useful: Theorem 1.5.1. Every group is an epimorphic image of a free group. Proof. Let G be a group, and let X be the underlying set of elements of G. Define FX to be the free group on X. Then the correspondence x 󳨃→ x (x ∈ X) uniquely defines an epimorphism FX → G. Lemma 1.5.2. Let G be a group and A a subgroup of G such that G/A = ⨁i∈I (Gi /A). If Gi = A ⊕ Bi (i ∈ I), then G = A ⊕ ⨁i∈I Bi . Proof. It is clear that we have G = A + ∑i∈I Bi . To show that the sum is direct, suppose that a + b1 + ⋅ ⋅ ⋅ + bn = 0 with elements a ∈ A and bk ∈ Bik (ik ∈ I, k = 1, . . . , n). Then (b1 + A) + ⋅ ⋅ ⋅ + (bn + A) = 0, which yields bk ∈ A ∩ Bik = 0 for all k = 1, . . . , n. It follows that b1 = ⋅ ⋅ ⋅ = bn = 0, which implies a = 0. Theorem 1.5.3. Let G be a group and A a subgroup of G. If G/A is free, then G = A ⊕ F for some free group F. Proof. We write G/A = ⨁i∈I (Gi /A) with Gi /A ≅ ℤ (i ∈ I), and let i ∈ I. Then Gi /A = ⟨xi + A⟩ for some xi ∈ Gi . Since this group is torsion-free, xi has infinite order, and A ∩ ⟨xi ⟩ = 0; thus, Gi = A ⊕ ⟨xi ⟩. Letting F = ⨁i∈I ⟨xi ⟩, we obtain G = A ⊕ F by Lemma 1.5.2. The following important result will be needed: Theorem 1.5.4. Every subgroup of a free group is free. Proof. Let F be a free group and A a subgroup of F. We write F = ⨁α αi for infinitely many i.

1.12 Proper elements and height-preserving homomorphisms Let A be a submodule of an R-module G and p ∈ R a prime. Then an element x ∈ G is called p-proper (or proper if p is understood) with respect to A if x has maximal

30 | 1 Basic characteristics of abelian groups p-height in the coset x + A; that is, if |x|p ≥ |x + a|p for all a ∈ A. Notice that such an element always exists in cases where A is finite. The following observation will be useful: Lemma 1.12.1. Let A be a submodule of an R-module G, p ∈ R a prime, x ∈ G and α an ordinal. Then we have the following: 1. x is proper with respect to A if and only if |x + a|p = |x|p ∧ |a|p for all a ∈ A. 2. Assume that x ∈ ̸ A and |x|p = α. Then x is proper with respect to A if and only if x ∉ A + pα+1 G. Proof. To verify (1), suppose that x is proper with respect to A and let a ∈ A. Then |a|p ≥ |x+a|p ∧|x|p ≥ |x+a|p ; hence, |x|p ∧|a|p ≥ |x+a|p . Therefore, we have |x|p ∧|a|p = |x+a|p . The converse is obvious. The second statement is clear. Lemma 1.12.2. Let A be a submodule of an R-module G, x ∈ G \ A, and p ∈ R a prime. Then the following conditions are equivalent: 1. The coset x + A contains an element which is proper with respect to A; 2. There exists an element y ∈ x + A such that |y|Gp = |x + A|G/A p ; α α 3. If α is an ordinal and x + A ∈ p (G/A), then x ∈ p G + A. Proof. (1) implies (3). Suppose that x + z is proper with respect to A, where z ∈ A. Let α > 0 be an ordinal and assume that the claim is true for all β < α. Suppose x +A ∈ pα (G/A). If α−1 exists, then there is a y+A ∈ pα−1 (G/A) such that x +A = p(y+A). By induction, we have y + A ∈ pα−1 G + A; hence, x ∈ pα G + A. If α is a limit ordinal, let β < α. Then x + A ∈ pβ (G/A); thus, x ∈ pβ G + A. Therefore, there exists a yβ ∈ A such that |x + yβ |p ≥ β. But then |x + z|p ≥ |x + yβ |p ≥ β; therefore, x ∈ pα G + A, as desired. (3) implies (2). Suppose that |x + A|p = α. Then from (3), there is a y ∈ x + A such that |y|p ≥ α. Therefore, |x + A|p = |y|p . Finally, suppose that (2) holds. Then (1) is immediate since |y|p = |y + A|p ≥ |x + a|p for all a ∈ A. Corollary 1.12.3. Let A be a submodule of an R-module G, x ∈ G \ A, and p ∈ R a prime. Then the following conditions are equivalent: 1. x is proper with respect to A; 2. |x + A|pG/A = |x|Gp . Proof. If x is proper with respect to A, then by Lemma 1.12.2(2), there is a y ∈ x + A such that |y|p = |x + A|p ; hence, |x|p ≥ |y|p = |x + A|p ≥ |x|p . The converse is obvious. Suppose that S and T are submodules of R-modules G and H, respectively, and let p ∈ R be a prime. Then we say that a homomorphism f : S → T does not decrease G H p-heights if |f (x)|H p ≥ |x|p for all x ∈ S, and f is called p-height-preserving if |f (x)|p = |x|Gp for all x ∈ S. If f is a p-height-preserving homomorphism for all primes p ∈ R, then we call f height-preserving. Notice that if g : G → H is an isomorphism, then the

1.13 Ulm invariants | 31

restriction g ↾ S : S → g(S) is a height-preserving isomorphism. Proper elements can be used to extend homomorphisms between submodules which preserve or do not decrease p-heights: Lemma 1.12.4. Let G and H be R-modules, p ∈ R a prime, and f : S → T a homomorphism, where S and T are submodules of G and H, respectively. Let a ∈ G \ S and b ∈ H such that pa ∈ S, pb ∈ T and f (pa) = pb. Then we have the following: 1. f extends to a homomorphism g : ⟨S, a⟩ → ⟨T, b⟩ by sending a onto b. 2. If f is an isomorphism and b ∉ T, then the map g in (1) is an isomorphism. 3. If f is p-height-preserving, |a|p = |b|p , a is proper with respect to S, and b is proper with respect to T, then the map g in (1) is p-height-preserving. 4. If f does not decrease p-heights, |a|p ≤ |b|p and a is proper with respect to S, then the map g in (1) does not decrease p-heights. Proof. Statements (1) and (2) were shown in Lemma 1.6.5 for R = ℤ; the proof for arbitrary R carries over. To verify (3), let z = s + ra, where s ∈ S and r ∈ R. If p|r, then z ∈ S and the p-heights |f (z)| and |z| coincide by assumption. If p ∤ r, then there are k, l ∈ R such that kp + lr = 1; hence, lz = l(ra + s) = a − kpa + ls ∈ a + S. Therefore, we may write lz = a + s󸀠 for some s󸀠 ∈ S. Since p ∤ l, we have |z| = |lz| = |a + s󸀠 | = |a| ∧ |s󸀠 | since a is proper with respect to S. Likewise, |g(z)| = |b| ∧ |f (s󸀠 )| = |a| ∧ |s󸀠 | since b is proper with respect to T, and we conclude that |g(z)| = |z|. A slight modification of the proof of (3) shows the last statement. Exercises. 1. If G and H are R-modules and ϕ : G → H is a monomorphism, then ϕ(pα G) = pα ϕ(G) for all primes p ∈ R and ordinals α. 2. If G is an R-module and G = A ⊕ B, then pα G = pα A ⊕ pα B for all primes p ∈ R and ordinals α. 3. Let G be an R-module, p ∈ R a prime, and α an ordinal. Then every coset x + pα G ∈ G/pα G has an element p-proper with respect to pα G.

1.13 Ulm invariants Ulm’s theorem, which will be proved in the next section, was one of the first theorems to classify infinite groups by defining invariants that completely determine the groups in some class up to isomorphism. Here we define the invariant, which has subsequently been applied to broader classes of groups, as we shall see. Let G be an R-module. Then for a given prime p ∈ R, the Ulm invariants of G are defined by up (α, G) = dim(pα G)[p]/(pα+1 G)[p]

32 | 1 Basic characteristics of abelian groups if α is an ordinal and up (∞, G) = dim(p∞ G)[p] with the groups considered as vector spaces over the field R/(p). The subscript p may be dropped if it is clear which prime is used. For instance, if G is a direct sum of cyclic p-groups and n < ω, then u(n, G) is the number of direct summands of order pn+1 , and we have u(α, G) = 0 for all α ≥ ω. Thus, two direct sums of cyclic p-groups are isomorphic exactly if they have the same Ulm invariants. Since the torsion-free rank of a group is an invariant (Corollary 1.4.3), we have the following: Corollary 1.13.1. The direct decomposition of a finitely generated group as in Theorem 1.9.3 is unique up to the order in which the cyclic direct summands of infinite and prime power order are written. Two finitely generated groups G and H are isomorphic if and only if r0 (G) = r0 (H) and up (n, G) = up (n, H) for all n < ω and primes p. The following hold for any group G and prime p: (a) If G = ⨁i∈I Gi , then up (α, G) = ∑i∈I up (α, Gi ) for all α ∈ Ord∞ . (b) If α is an ordinal, then rp (pα G) ≥ ∑γ≥α up (γ, G). Remark 1.13.2. If G is any group and p is a prime, then tp G = D ⊕ A with D divisible and A reduced (Theorem 1.6.4); thus, (a) yields up (α, G) = up (α, A) for all ordinals α. Next we wish to establish a connection between Ulm invariants of a given group and any of its p-basic subgroups. The following lemma is needed (see [29, p. 168f]): Lemma 1.13.3. Suppose that B = ⨁n α + 1, contradicting our choice of a. Notice that b ∉ T since |b| ≠ ∞. By Lemma 1.12.4, f can be extended to a p-height-preserving isomorphism g : ⟨S, a⟩ → ⟨T, b⟩ by sending a onto b.

36 | 1 Basic characteristics of abelian groups Case II: |pa| > α + 1. Then f (pa) = pb󸀠 for some b󸀠 ∈ pα+1 H. We write pa = pa󸀠 with a󸀠 ∈ pα+1 G and have a − a󸀠 ∈ (pα G)[p]. Since a is proper, a − a󸀠 ∉ S + pα+1 G, and it follows that uα,p (H, T) = uα,p (G, S) = dim

(pα G)[p] ≠ 0. ∩ (S + pα+1 G)

(pα G)[p]

Therefore there is a b̃ ∈ (pα H)[p] with b̃ ∉ T + pα+1 H. Letting b = b󸀠 + b,̃ we obtain b ∉ T since b󸀠 ∈ pα+1 H. Using the fact that b̃ is proper with respect to T, we obtain |b| = α = |b|̃ ≥ |b̃ + z| = |b󸀠 + b̃ + z| = |b + z| for all z ∈ T. Consequently, b is proper with respect to T. In addition, we have pb = pb󸀠 + pb̃ = f (pa), so again by Lemma 1.12.4, f extends to the desired map g. Finally, suppose that |a| = ∞. Then |f (pa)| = ∞; hence, f (pa) = pb󸀠 for some 󸀠 b ∈ p∞ H. In view of Lemma 1.12.4, it suffices to show that there is a b ∈ H \ T such that f (pa) = pb and |b| = ∞. If b󸀠 ∉ T, we let b = b󸀠 and are done. Therefore, suppose that b󸀠 ∈ T. Then b󸀠 = f (y) for some y ∈ S; hence, f (pa − py) = 0. But then p(a − y) = 0. Thus, a − y ∈ (p∞ G)[p]. Since a − y ∉ S, we obtain u∞,p (H, T) = u∞,p (G, S) = dim

(p∞ G)[p] ≠ 0. (p∞ G ∩ S)[p]

Therefore, there is a b̃ ∈ (p∞ H)[p] \ (p∞ H ∩ T)[p]. Letting b = b󸀠 + b,̃ we have b ∉ T, |b| = ∞, and pb = pb󸀠 + pb̃ = f (pa), as desired. Now we are ready to prove the following Ulm’s theorem. Theorem 1.14.2 (Ulm [146]). Two countable p-groups G and H are isomorphic if and only if up (α, G) = up (α, H) for all α ∈ Ord∞ . Proof. Suppose that G and H have the same Ulm invariants. Let I be the set of all height-preserving isomorphisms f : S → T, where S and T are finite subgroups of G and H, respectively. We will show that I has the back-and-forth property. Clearly, I is nonempty. Therefore, let f : S → T be an element of I and assume that a ∈ G \ S. Since G is a p-group, we may assume that pa ∈ S. In addition, we can assume that a is proper with respect to S since S is finite. Notice that for any α, an ordinal or ∞, we have Iα (S) ≅ Iα (T) by Lemma 1.13.9. Therefore, since Iα (S) is finite, we obtain uα (G, S) = uα (H, T) by Lemma 1.13.8. Then Lemma 1.14.1 shows that f extends to a map g ∈ I with a ∈ domain(g). By symmetry, the set I has the back-and-forth property. Since G and H are countable groups, this yields a step-by-step construction of an isomorphism G → H. The converse of the statement is immediate. In view of Theorem 1.3.2, we obtain the following: Corollary 1.14.3. Two countable torsion groups are isomorphic if and only if they have the same Ulm invariants up (p ∈ ℙ).

1.15 Completely decomposable torsion-free groups | 37

Exercises. 1. (Zippin [157]) Let G and H be countable p-groups having the same Ulm invariants. If α is an ordinal, then any isomorphism pα G → pα H extends to an isomorphism G → H. 2. Give an example of p-groups G and H with the same Ulm invariants such that n G ≇ H. [Hint: Let G be the torsion part of ∏∞ n=1 ℤ/p ℤ.]

1.15 Completely decomposable torsion-free groups Now that we have a classification theorem for a class of torsion groups, we turn to torsion-free groups and Baer’s classification theorem. We start with a basic observation on p-heights of elements in a torsion-free group G. Lemma 1.15.1. Let G be a torsion-free group, x ∈ G, and p a prime. Then we have the following: 1. |x|p < ω or |x|p = ∞. 2. |px|p = |x|p + 1 (we set ∞ + 1 = ∞). Proof. Suppose that x ∈ pω G. For every n < ω, there is a yn ∈ G such that x = pn+1 yn . Then p(y0 − pn yn ) = 0, which implies y0 = pn yn since G is torsion-free. It follows that y0 ∈ pω G; hence, x = py0 ∈ pω+1 G. Thus, pω G = pω+1 G, and we have |x|p = ∞. To show (2), let n < ω. If px ∈ pn+1 G, then px = p(pn y) for some y ∈ G; thus, x = pn y ∈ pn G. It follows that |px|p = |x|p + 1. It follows that the Ulm matrix of an element x ∈ G is completely determined by the sequence of its p-heights χG (x) = (|x|pi )i∈ℕ , called the characteristic of x. More generally, by a characteristic we mean an ordered sequence (h1 , . . . , hi , . . . ) of nonnegative integers and symbols ∞. We write (h1 , . . . , hi , . . . ) ≥ (k1 , . . . , ki , . . . ) whenever hi ≥ ki for all i ∈ ℕ. Two characteristics (h1 , . . . , hi , . . . ) and (k1 , . . . , ki , . . . ) are called equivalent if ∑i∈ℕ |hi − ki | is finite (we set ∞ − ∞ = 0). In other words, the sequences are equivalent if and only if hi ≠ ki only for finitely many i, and for such i, both hi and ki are finite. The resulting equivalence classes are called types. The type of an element x ∈ G, written t(x), is the type containing χG (x). The typeset of G is defined to be the set typeset(G) = {t(x) : 0 ≠ x ∈ G}. If every nonzero element of G has the same type t, then we say that G has type t and write t(G) = t. In this case, G is called homogeneous. For example, torsion-free rank 1 groups (= the nontrivial subgroups of ℚ; see Theorems 1.6.8, 1.6.9 and 1.6.7) are homogeneous.

38 | 1 Basic characteristics of abelian groups For types s and t, we write s ≥ t if (h1 , . . . , hi , . . . ) ≥ (k1 , . . . , ki , . . . ) for some elements (h1 , . . . , hi , . . . ) ∈ s and (k1 , . . . , ki , . . . ) ∈ t. By s∧t, we mean the type of the characteristic (h1 ∧ k1 , . . . , hi ∧ ki , . . . ). For G a torsion-free group, we have the following: (a) If x, y ∈ G, then mx = ny for some nonzero integers m, n implies t(x) = t(y). (b) If x, y ∈ G, then t(x + y) ≥ t(x) ∧ t(y). (c) If G = A ⊕ B and x ∈ A, y ∈ B, then t(x + y) = t(x) ∧ t(y). (d) If α : G → H is a homomorphism with H torsion-free, then t(x) ≤ t(α(x)) for all x ∈ G. For a torsion-free group G and a type t, we define G(t) = {x ∈ G : t(x) ≥ t} and G∗ (t) = ⟨x ∈ G : t(x) > t⟩. Then the Baer invariants of G are defined to be the torsion-free ranks of the factor groups Gt = G(t)/G∗ (t) for all types t. Lemma 1.15.2. Let G, H be torsion-free rank 1 groups and assume that g ∈ G, h ∈ H are nonzero elements such that χG (g) ≤ χH (h). Then the correspondence g 󳨃→ h extends to a monomorphism α : G → H. The map α is an isomorphism if χG (g) = χH (h). Proof. Given integers m ≠ 0 and n, the equation my = nh is solvable in H if the corresponding equation mx = ng is solvable in G, and any such solutions are unique since the groups are torsion-free. Thus, g 󳨃→ h extends to a homomorphism α : G → H. To show that α is injective, let z ∈ ker α. Then mg = nz for some integers m and n ≠ 0, and we have mα(g) = α(mg) = α(nz) = nα(z) = 0. But then m = 0 since α(g) = h ≠ 0. It follows that z = 0; thus, α is injective. In the case χG (g) = χH (h), in the first sentence of this proof, “if” can be replaced by “if and only if”; hence, α is bijective. Theorem 1.15.3 (Baer [6]). Two torsion-free rank 1 groups G and H are isomorphic if and only if they have the same type. For every type t, there is a subgroup A of ℚ which has type t. Proof. Suppose that t(G) = t(H) and let g1 ∈ G, h1 ∈ H be nonzero elements. Then t(g1 ) = t(h1 ); hence, |g1 |p = |h1 |p for all but finitely many primes p, and |g1 |p ≠ ∞ exactly if |h1 |p ≠ ∞. Therefore, there are positive integers r and s such that g = rg1 and h = sh1 satisfy χG (g) = χH (h). By Lemma 1.15.2, the correspondence g 󳨃→ h extends to an isomorphism G → H.

1.15 Completely decomposable torsion-free groups | 39

If t = [(h1 , . . . , hi , . . . )] is a type, then let A be the subgroup of ℚ generated by the −r set {pi i : 0 ≤ ri ≤ hi (i ∈ ℕ)}. Then χA (1) = (h1 , . . . , hi , . . . ), and we have t(A) = t.

A torsion-free group is called completely decomposable if it is a direct sum of rank 1 groups. Suppose G is completely decomposable, say G = ⨁i∈I Gi , where each group Gi has rank 1. For any type t, we define the sets It = {i ∈ I : t(Gi ) ≥ t} and Jt = {i ∈ I : t(Gi ) > t}. Note that (xi )i∈I ∈ G(t) if and only if xi ∈ Gi (t) for all i ∈ I. Then we have G(t) = ⨁i∈It Gi and G∗ (t) = ⨁i∈Jt Gi by (c); hence, the rank of Gt is equal to |{i ∈ I : t(Gi ) = t}|. Since Gt does not depend on any particular decomposition of G, we obtain the following: Theorem 1.15.4 (Baer [6]). Let G be a completely decomposable group. If G = ⨁i∈I Gi = ⨁j∈J Hj such that each group Gi and Hj has rank 1, then there is a bijection ϕ : I → J such that Gi ≅ Hϕ(i) for all i ∈ I. Two completely decomposable groups are isomorphic if and only if they have the same Baer invariants. We state the following important fact, which will follow from a more general result (cf. Corollary 3.16.5): Theorem 1.15.5 (Baer [6], Kulikov [90], Kaplansky [79]). The class of completely decomposable groups is closed under direct summands. The following fact will be needed later: Proposition 1.15.6. Let G and H be torsion-free rank 1 groups. Then the following conditions are equivalent: 1. There is a monomorphism G → H; 2. There is a nonzero homomorphism G → H; 3. t(G) ≤ t(H). Proof. Clearly (1) implies (2). Given any homomorphism α : G → H and x ∈ G, we have χG (x) ≤ χH (α(x)), thus (2) implies (3). Assuming that (3) is true, there are nonzero elements g ∈ G and h ∈ H such that χG (g) ≤ χH (h). By Lemma 1.15.2, the correspondence g 󳨃→ h extends to a monomorphism G → H. Thus, (1) follows.

Exercises. 1. For a subgroup A of a torsion-free group G, define A∗ to be the intersection of all pure subgroups of G containing A, called the purification of A. Show the following: (a) A∗ is a pure subgroup of G. (b) A∗ = A0 (see Section 1.10). (c) A∗ /A = t(G/A). 2. If G is torsion-free and G = A ⊕ B, then G(t) = A(t) ⊕ B(t) and G∗ (t) = A∗ (t) ⊕ B∗ (t) for all types t.

40 | 1 Basic characteristics of abelian groups 3.

Let G and H be torsion-free rank 1 groups. Then G ≅ H if and only if there is a homomorphism G → H whose cokernel is bounded. 4. Give an example of a completely decomposable group having a pure subgroup that is not completely decomposable.

1.16 Hom and Ext Given two groups we may use them to form other useful groups, specifically the group of homomorphisms and the group of extensions. For homomorphisms α, β : A → C, we define their sum to be the map α+β:

A x

→ 󳨃→

C α(x) + β(x)

and obtain the group of homomorphisms from A to C, which is denoted by Hom(A, C). Notice that for A = ℤ, we have an isomorphism ∼

Hom(ℤ, C) 󳨀→ C given by ϕ 󳨃→ ϕ(1). Likewise, we obtain an isomorphism ∼

Hom(ℤ/nℤ, C) 󳨀→ C[n]. The following items are easily verified: (a) If C is torsion-free, then so is Hom(A, C). (b) If A is torsion and C is torsion-free, then Hom(A, C) = 0. (c) If A is divisible, then Hom(A, C) is torsion-free. (d) If A is torsion-free and divisible, then so is Hom(A, C). It is clear that a given group G and α ∈ Hom(A, B) give rise to homomorphisms α∗ :

Hom(B, G) ϕ

→ 󳨃→

Hom(A, G) ϕα

α∗ :

Hom(G, A) ϕ

→ 󳨃→

Hom(G, B) αϕ.

and

Of particular note is when α is multiplication by a positive integer. The following simple—but useful—observation shows that we may look at such a multiplication either as a homomorphism or as the result of repeated additions: Proposition 1.16.1. Multiplication by a positive integer n on A or C induces multiplication by n on Hom(A, C); that is, (n1A )∗ = (n1C )∗ = n1Hom(A,C) .

1.16 Hom and Ext | 41

Proof. Consider the map α : A → A given by x 󳨃→ nx. If ϕ ∈ Hom(A, C), then ϕα(x) = ϕ(nx) = nϕ(x); thus, α∗ : ϕ 󳨃→ nϕ. Likewise, the map γ : C → C with γ : y 󳨃→ ny induces γ∗ : ϕ 󳨃→ nϕ. Exactness of a short exact sequence induces exactness of Hom sequences, as shown in the following: Theorem 1.16.2 (Cartan–Eilenberg [12]). Given any group G, exactness of the sequence α

β

E : 0 → A 󳨀→ B 󳨀→ C → 0 induces exactness of the sequences β∗

α∗

α∗

β∗

0 → Hom(C, G) 󳨀→ Hom(B, G) 󳨀→ Hom(A, G) and 0 → Hom(G, A) 󳨀→ Hom(G, B) 󳨀→ Hom(G, C). Proof. To show that β∗ is injective, assume that ϕ ∈ ker β∗ . Then ϕβ = 0 implies that ϕ = 0 since β is surjective; therefore, β∗ is injective. For any ϕ ∈ Hom(C, G), we have α∗ β∗ (ϕ) = ϕβα = 0; thus, im β∗ ⊆ ker α∗ . To prove the reverse inclusion, let ϕ ∈ ker α∗ . Then ϕα = 0, so by Lemma 1.2.2 there is a ν ∈ Hom(C, G) with νβ = ϕ; thus, ϕ ∈ imβ∗ . It follows that the first sequence is exact. Letting ϕ ∈ ker α∗ , we obtain αϕ = 0; thus, ϕ = 0 since α is injective. Therefore, α∗ is injective. If ϕ ∈ Hom(G, A), then β∗ α∗ (ϕ) = βαϕ = 0, which shows that im α∗ ⊆ ker β∗ . For the reverse inclusion, let ϕ ∈ ker β∗ . Then βϕ = 0, so by Lemma 1.2.1 there is a ν ∈ Hom(G, A) such that αν = ϕ; thus, ϕ ∈ imα. Now let A, Ai , C, Ci (i ∈ I) be groups. If α : ⨁i Ai → C is a homomorphism and αi = α ↾ Ai (i ∈ I), then α 󳨃→ (αi )i∈I defines a map from Hom(⨁i Ai , C) to ∏i Hom(Ai , C). A straightforward calculation shows that this map is an isomorphism. Likewise, given a homomorphism β : A → ∏i Ci and projection maps πi : ∏i Ci → Ci , the correspondence β 󳨃→ (πi β)i∈I defines an isomorphism between Hom(A, ∏i Ci ) and ∏i Hom(A, Ci ). Thus, we have the following: Theorem 1.16.3. There are natural isomorphisms Hom(⨁ Ai , C) ≅ ∏ Hom(Ai , C) i∈I

i∈I

and Hom(A, ∏ Ci ) ≅ ∏ Hom(A, Ci ). i∈I

i∈I

Now we turn our attention to the group of extensions. We will define an equivalence relation and an addition that make it an abelian group. As we did with Hom, we will define morphisms between groups of extensions based on homomorphisms on

42 | 1 Basic characteristics of abelian groups the underlying groups. Finally, we will use these groups to extend the exact sequence of Theorem 1.16.2 above. A short exact sequence of groups μ

ν

E : 0 → A 󳨀→ B 󳨀→ C → 0 is called an extension of A by C. By a morphism between the extensions E and E 󸀠 : μ󸀠

ν󸀠

0 → A󸀠 󳨀→ B󸀠 󳨀→ C 󸀠 → 0, we mean a triple Γ = (α, β, γ) of homomorphisms such that the diagram μ

0

→

A ↓α

󳨀→

0

→

A󸀠

󳨀→

μ󸀠

ν

B ↓β

󳨀→

B󸀠

󳨀→

ν󸀠

C ↓γ

→

0

C󸀠

→

0

is commutative, and we write Γ : E → E 󸀠 . If Γ = (1A , β, 1C ) for some β : B → B󸀠 , then we call E and E 󸀠 equivalent, written E ≡ E 󸀠 . In this case, the map β is automatically an isomorphism by Lemma 1.2.3; thus, ≡ is an equivalence relation. Then Ext(C, A) denotes the set of all equivalence classes of extensions of A by C. We say that the extension E splits if E is equivalent to the split extension E0 : 0 → A → A ⊕ C → C → 0. For instance, E splits whenever A is divisible or C is free (Theorems 1.6.3 and 1.5.3). Given the extension E and a homomorphism α : A → A󸀠 , we obtain a pushout diagram, by which we mean the commutative diagram with exact rows E:

0

→

αE :

0

→

μ

A ↓α

B ↓β

󳨀→ σ

A󸀠

X

󳨀→

ν

󳨀→ τ

󳨀→

C ‖

C

→

0

→

0,

such that X = (A󸀠 ⊕ B)/L with L = {(−α(a), μ(a)) : a ∈ A} and σ : g 󳨃→ (g, 0) + L, τ : (g, b) + L 󳨃→ ν(b), β : b 󳨃→ (0, b) + L. Note that any commutative diagram with exact rows μ

E:

0

→

A ↓α

󳨀→

E󸀠 :

0

→

A󸀠

󳨀→

μ󸀠

ν

B ↓ β󸀠

󳨀→

→

0

ν󸀠

C ↓ γ󸀠

B󸀠

󳨀→

C󸀠

→

0

B ↓β

󳨀→

ν

C ‖

→

0

C ↓ γ󸀠

→

0

C󸀠

→

0

gives rise to the commutative diagram A ↓α

μ

E:

0

→

αE :

0

→

A󸀠 ‖

󳨀→

E󸀠 :

0

→

A󸀠

󳨀→

󳨀→ σ

μ󸀠

τ

X ↓f

󳨀→

B󸀠

󳨀→

ν󸀠

1.16 Hom and Ext | 43

with f : (a󸀠 , b) + L 󳨃→ μ󸀠 (a󸀠 ) + β󸀠 (b), and we obtain fβ = β󸀠 . Following Mac Lane’s [111] terminology, we say that Γ󸀠 = (α, β󸀠 , γ 󸀠 ) : E → E 󸀠 can be factored through Γ = (α, β, 1C ) : E → αE; that is, Γ󸀠 can be written as a composite Γ

E 󳨀→ αE

(1A󸀠 ,f ,γ󸀠 ) 󸀠 󳨀→ E .

In particular, any commutative diagram E:

0

→

Ẽ :

0

→

A ↓α A󸀠

B ↓ B̃

→ →

C ‖ C

→ →

→

0

→

0

yields αE ≡ E.̃ Thus, we have the following: (i) 1A E ≡ E. (ii) If E ≡ E 󸀠 , then αE ≡ αE 󸀠 for any α ∈ Hom(A, A󸀠 ). (iii) (α󸀠 α)E ≡ α󸀠 (αE) for any α ∈ Hom(A, A󸀠 ) and α󸀠 ∈ Hom(A󸀠 , A󸀠󸀠 ). Then (ii) shows that for any [E] ∈ Ext(C, A) and α ∈ Hom(A, A󸀠 ), there is a uniquely defined [αE] ∈ Ext(C, A󸀠 ). Dually, a homomorphism γ : C 󸀠 → C gives rise to the commuting diagram with exact rows Eγ :

0

→

E:

0

→

ϕ

A ‖

μ

A

ψ

Y ↓η

󳨀→

ν

B

󳨀→

C󸀠 ↓γ

󳨀→

C

󳨀→

→

0

→

0,

which is called a pullback diagram, where Y = {(b, c󸀠 ) ∈ B ⊕ C 󸀠 : ν(b) = γ(c󸀠 )} and ϕ : a 󳨃→ (μ(a), 0), ψ : (b, c󸀠 ) 󳨃→ c󸀠 , η : (b, c󸀠 ) 󳨃→ b. Then any morphism Γ1 = (α1 , β1 , γ) : E 1 → E can be factored through Γ = (1A , η, γ) : Eγ → E; that is, Γ1 is a composite Γ

(α1 ,g,1C )

E 1 󳨀→ Eγ 󳨀→ E for some map g. To see this, assume that E1 :

0

→

E:

0

→

A1 ↓ α1 A

ϕ1

󳨀→ μ

󳨀→

B1 ↓ β1 B

ψ1

󳨀→ ν

󳨀→

C󸀠 ↓γ C

→

0

→

0

is a commutative diagram with exact rows. Then we obtain the commutative diagram ϕ1

E1 :

0

→

A1 ↓ α1

󳨀→

Eγ :

0

→

󳨀→

E:

0

A ‖

→

A

ϕ μ

󳨀→

ψ1

B1 ↓g

󳨀→

Y ↓η

󳨀→

B

ψ

ν

󳨀→

C󸀠 ‖

→

0

C󸀠 ↓γ

→

0

→

0

C

44 | 1 Basic characteristics of abelian groups by letting g : b1 󳨃→ (β1 (b1 ), ψ1 (b1 )), and we have ηg = β1 , proving the assertion. Therefore, any commutative diagram with exact rows E 󸀠󸀠 :

0

→

E:

0

→

A ‖

A

B󸀠󸀠 ↓

→ μ

ν

B

󳨀→

C󸀠 ↓γ

→

C

󳨀→

→

0

→

0

yields Eγ ≡ E 󸀠󸀠 . This shows the following: (iv) E1C ≡ E. (v) If E ≡ E 󸀠 , then Eγ ≡ E 󸀠 γ for any γ ∈ Hom(C 󸀠 , C). (vi) E(γγ 󸀠 ) ≡ (Eγ)γ 󸀠 whenever γ 󸀠 ∈ Hom(C 󸀠󸀠 , C 󸀠 ) and γ ∈ Hom(C 󸀠 , C). Clearly, the diagram Eγ :

0

→

αE :

0

→

A ↓α

A󸀠

ϕ

Y ↓ βη

󳨀→ σ

X

󳨀→

ψ

C󸀠 ↓γ

󳨀→ τ

C

󳨀→

→

0

→

0

1

commutes, so by the observation above (with E and E replaced by Eγ and αE, respectively), we obtain a commutative diagram Eγ :

0

→

(αE)γ :

0

→

ϕ

A ↓ A󸀠

󳨀→ →

Y ↓ Z

ψ

C󸀠 ‖ C󸀠

󳨀→ →

→

0

→

0.

On the other hand, Eγ and α : A → A󸀠 generate a pushout diagram; thus, we obtain the equivalence α(Eγ) ≡ (αE)γ. Proposition 1.16.4. If μ

E:

0

→

A ↓α

󳨀→

E󸀠 :

0

→

A󸀠

󳨀→

μ󸀠

ν

B ↓β

󳨀→

B󸀠

󳨀→

ν󸀠

C ↓γ

→

0

C󸀠

→

0

is a commutative diagram with exact rows, then αE ≡ E γ. 󸀠

Proof. Factoring Γ = (α, β, γ) : E → E 󸀠 through Γ̃ : E → αE, we write Γ as a composΓ̃

(1A󸀠 ,f ,γ)

ite E 󳨀→ αE 󳨀→ E 󸀠 for some suitable map f . Since E 󸀠 and γ generate a pullback diagram, we obtain αE ≡ E 󸀠 γ. Recall that the zero map on a group A is denoted by 0A . We have the following basic fact: Lemma 1.16.5. Given an extension E : 0 → A → B → C → 0, we have 0A E ≡ E0 and E0C ≡ E0 , where E0 is the split extension.

1.16 Hom and Ext | 45

Proof. The assertion follows from the commutativity of the diagram E0 :

0

→

E:

0

→

E0 :

0

→

A ‖

A⊕C ↓α

→ μ

A ↓ 0A A

B ↓β A⊕C

󳨀→ →

→ ν

󳨀→ →

C ↓ 0C C ‖ C

→

0

→

0

→

0

with α : (a, c) 󳨃→ μ(a) and β : b 󳨃→ (0, ν(b)). Next we will show that the set Ext(C, A) can be turned into an abelian group using the diagonal map ΔG :

G x

→ 󳨃→

G⊕G (x, x)

and the codiagonal map ∇G :

G⊕G (x, y)

→ 󳨃→

G x + y.

Let ϕG : G ⊕ (G ⊕ G) → (G ⊕ G) ⊕ G denote the natural isomorphism and define the direct sum of homomorphisms σi : Gi → Hi (i = 1, 2) to be σ1 ⊕ σ2 :

G1 ⊕ G2 (x1 , x2 )

→ 󳨃→

H1 ⊕ H2 (σ1 (x1 ), σ2 (x2 )).

The following identities involving the maps above are easily verified: (a) (ΔG ⊕ 1G )ΔG = ϕG (1G ⊕ ΔG )ΔG . (b) ∇G (∇G ⊕ 1G ) = ∇G (1G ⊕ ∇G )ϕ−1 G . (c) If σ1 , σ2 ∈ Hom(C, A), then σ1 + σ2 = ∇A (σ1 ⊕ σ2 )ΔC . (d) If α ∈ Hom(A, A󸀠 ), then α∇A = ∇A󸀠 (α ⊕ α). (e) If γ ∈ Hom(C 󸀠 , C), then ΔC γ = (γ ⊕ γ)ΔC󸀠 . μi

νi

The direct sum of extensions Ei : 0 → Ai 󳨀→ Bi 󳨀→ Ci → 0 (i = 1, 2) is defined to be the induced extension μ1 ⊕μ2

ν1 ⊕ν2

E1 ⊕ E2 : 0 → A1 ⊕ A2 󳨀→ B1 ⊕ B2 󳨀→ C1 ⊕ C2 → 0. Letting τG :

G1 ⊕ G2 (x1 , x2 )

→ 󳨃→

G2 ⊕ G1 (x2 , x1 ),

we obtain a morphism (τA , τB , τC ) : E1 ⊕ E2 → E2 ⊕ E1 . We have the following:

46 | 1 Basic characteristics of abelian groups (f) ∇A τA = ∇A and ΔC = τC ΔC . (g) If αi ∈ Hom(Ai , A󸀠i ) (i = 1, 2), then (α1 ⊕ α2 )(E1 ⊕ E2 ) ≡ α1 E1 ⊕ α2 E2 . (h) If γi ∈ Hom(Ci󸀠 , Ci ) (i = 1, 2), then (E1 ⊕ E2 )(γ1 ⊕ γ2 ) ≡ E1 γ1 ⊕ E2 γ2 . The sum of two extensions of A by C as defined in the next theorem is known as the Baer sum. Note the similarity to (c) above. Theorem 1.16.6 (Mac Lane [111]). Let A and C be groups. Then the set Ext(C, A) is an abelian group under the operation defined by [E1 ] + [E2 ] = [E1 + E2 ], where E1 + E2 = ∇A (E1 ⊕ E2 )ΔC with extensions E1 and E2 of A by C. The identity of Ext(C, A) is the equivalence class of the split extension E0 : 0 → A → A⊕C → C → 0 and the inverse of [E] is the equivalence class of (−1A )E. If α, αi ∈ Hom(A, A󸀠 ) and γ, γi ∈ Hom(C 󸀠 , C) (i = 1, 2), then 1. α(E1 + E2 ) ≡ αE1 + αE2 , (E1 + E2 )γ ≡ E1 γ + E2 γ, 2. (α1 + α2 )E ≡ α1 E + α2 E, E(γ1 + γ2 ) ≡ Eγ1 + Eγ2 . Proof. Let E and Ei be extensions of A by C (i = 1, 2, 3). We will start by proving (1) and (2). The equivalences in (1) are evident since α∇A (E1 ⊕ E2 )ΔC = ∇A󸀠 (α ⊕ α)(E1 ⊕ E2 )ΔC ≡ ∇A󸀠 (αE1 ⊕ αE2 )ΔC and ∇A (E1 ⊕ E2 )ΔC γ = ∇A (E1 ⊕ E2 )(γ ⊕ γ)ΔC󸀠 ≡ ∇A (E1 γ ⊕ E2 γ)ΔC󸀠 . By Proposition 1.16.4, (ΔA , ΔB , ΔC ) : E → E ⊕ E yields ΔA E ≡ (E ⊕ E)ΔC , and thus, we have (α1 +α2 )E = ∇A󸀠 (α1 ⊕α2 )ΔA E ≡ ∇A󸀠 (α1 ⊕α2 )(E ⊕E)ΔC ≡ ∇A󸀠 (α1 E ⊕α2 E)ΔC = α1 E +α2 E. Likewise, (∇A , ∇B , ∇C ) : E ⊕ E → E yields E∇C ≡ ∇A (E ⊕ E); therefore, E(γ1 + γ2 ) = E∇C (γ1 ⊕ γ2 )ΔC󸀠 ≡ ∇A (E ⊕ E)(γ1 ⊕ γ2 )ΔC󸀠 ≡ ∇A (Eγ1 ⊕ Eγ2 )ΔC󸀠 = Eγ1 + Eγ2 . This proves (2). It remains to show that the Baer sum defines an abelian group structure on Ext(C, A). If E1󸀠 ≡ E1 and E2󸀠 ≡ E2 , we obtain E1 ⊕ E2 ≡ E1󸀠 ⊕ E2󸀠 ; thus, (E1 ⊕ E2 )ΔC ≡ (E1󸀠 ⊕ E2󸀠 )ΔC . Therefore, E1 + E2 ≡ E1󸀠 + E2󸀠 . Hence, our addition on Ext(C, A) is welldefined. To verify the associative law, we compute (E1 + E2 ) + E3 = ∇A (∇A (E1 ⊕ E2 )ΔC ⊕ E3 )ΔC

= ∇A (∇A ⊕ 1A )[(E1 ⊕ E2 ) ⊕ E3 ](ΔC ⊕ 1C )ΔC

= ∇A (1A ⊕ ∇A )ϕ−1 A [(E1 ⊕ E2 ) ⊕ E3 ]ϕC (1C ⊕ ΔC )ΔC , which is equivalent to ∇A (1A ⊕ ∇A )[E1 ⊕ (E2 ⊕ E3 )](1C ⊕ ΔC )ΔC by Proposition 1.16.4 since we have a morphism (ϕA , ϕB , ϕC ) : E1 ⊕ (E2 ⊕ E3 ) → (E1 ⊕ E2 ) ⊕ E3 . Thus, (E1 + E2 ) + E3 ≡ E1 + (E2 + E3 ). To prove the commutative law, we compute ∇A (E1 ⊕ E2 )ΔC = ∇A τA (E1 ⊕ E2 )ΔC ≡ ∇A (E2 ⊕ E1 )τC ΔC = ∇A (E2 ⊕ E1 )ΔC ,

1.16 Hom and Ext | 47

which shows that E1 +E2 ≡ E2 +E1 . To prove that [E0 ] is the identity, we use Lemma 1.16.5 and the first equivalence in (2) to obtain E + E0 ≡ 1A E + 0A E ≡ (1A + 0A )E = 1A E ≡ E, as required. Finally, we have E + (−1A )E ≡ 1A E + (−1A )E ≡ (1A − 1A )E = 0A E ≡ E0 ; thus, [(−1A )E] is the inverse of [E]. In the following, we often write E ∈ Ext(C, A) instead of [E] ∈ Ext(C, A). By Theorem 1.16.6(1), any homomorphisms α : A → G and γ : G → C induce natural homomorphisms α∗ :

Ext(C, A) E

→ 󳨃→

Ext(C, G) αE

γ∗ :

Ext(C, A) E

→ 󳨃 →

Ext(G, A) Eγ.

and

Then (iii) and (vi) yield (α󸀠 α)∗ = α∗ α∗󸀠

for all α ∈ Hom(A, A󸀠 ) and α󸀠 ∈ Hom(A󸀠 , A󸀠󸀠 ).

(γγ 󸀠 )∗ = γ 󸀠 ∗ γ ∗

for all γ 󸀠 ∈ Hom(C 󸀠󸀠 , C 󸀠 ) and γ ∈ Hom(C 󸀠 , C).

Theorem 1.16.6(2) can be rephrased as follows: Lemma 1.16.7. Let αi ∈ Hom(A, A󸀠 ) and γi ∈ Hom(C 󸀠 , C) for i = 1, 2. Then (α1 + α2 )∗ = (α1 )∗ + (α2 )∗ and (γ1 + γ2 )∗ = (γ1 )∗ + (γ2 )∗ . Therefore, leaving E fixed, we obtain homomorphisms E∗ :

Hom(A, G) σ

→ 󳨃→

Ext(C, G) σE

E∗ :

Hom(G, C) σ

→ 󳨃 →

Ext(G, A) Eσ,

and

which can be used to extend the Hom sequences in Theorem 1.16.2. By the next result, the newly obtained sequences E∗

Hom(B, G) → Hom(A, G) 󳨀→ Ext(C, G) and E∗

Hom(G, B) → Hom(G, C) 󳨀→ Ext(G, A) are exact: α

β

Lemma 1.16.8 (Mac Lane [111]). Let E : 0 → A 󳨀→ B 󳨀→ C → 0 be an exact sequence. Then we have the following:

48 | 1 Basic characteristics of abelian groups 1.

Given σ : A → G, the extension σE splits if and only if there is a map τ : B → G such that the diagram 0

2.

→

α

A σ ↓ G

β

B

󳨀→ ↙τ

󳨀→

C

→

0

is commutative. In particular, αE splits. Given σ : G → C, the extension Eσ splits if and only if there is a map τ : G → B such that the diagram

0

→

τ

α

A

B

󳨀→

↙

G ↓σ

󳨀→

C

β

→

0

is commutative. In particular, Eβ splits. Proof. (1): If there is such a map τ : B → G, then we have a commutative diagram α

E:

0

→

A ↓σ

󳨀→

E0 :

0

→

G

󳨀→

μ

β

B ↓ν

󳨀→

G⊕C

󳨀→

ϕ

C ‖

→

0

C

→

0

with ν : b 󳨃→ (τ(b), β(b)), μ : g 󳨃→ (g, 0) and ϕ : (g, c) 󳨃→ c; thus, σE ≡ E0 . Conversely, (σ,ν󸀠 ,1C )

(1G ,ν󸀠󸀠 ,1C )

assume that σE ≡ E0 . Then we have a composite E 󳨀→ σE 󳨀→ E0 and define τ = πν󸀠󸀠 ν󸀠 : B → G with π : G ⊕ C → G the projection map. We consider the resulting diagram and obtain τα = πμσ = σ, as required. (2): Assuming that there is a map σ : G → B, making the diagram in (2) commutative, we have μ

E0 :

0

→

A ‖

󳨀→

E:

0

→

A

󳨀→

α

ϕ

A⊕G ↓ν

󳨀→

B

󳨀→

β

G ↓σ

→

0

C

→

0

with ν : (a, g) 󳨃→ α(a) + τ(g), μ : a 󳨃→ (a, 0) and ϕ : (a, g) 󳨃→ g; hence, Eσ ≡ E0 . (1A ,ν󸀠 ,1C )

(1A ,ν󸀠󸀠 ,σ)

Conversely, suppose that Eσ ≡ E0 . Then there is a composite E0 󳨀→ Eσ 󳨀→ E. We define τ = ν󸀠󸀠 ν󸀠 ι with ι : G → A ⊕ G the inclusion map and obtain βτ = σϕι = σ. α

β

Corollary 1.16.9. Let E : 0 → A 󳨀→ B 󳨀→ C → 0 be an exact sequence. Then the following statements are equivalent: 1. E splits; 2. There is a map τ : B → A such that τα = 1A ; 3. There is a map τ : C → B such that βτ = 1C .

1.16 Hom and Ext | 49

Now we are ready to prove the following important result: Theorem 1.16.10 (Cartan–Eilenberg [12]). Given any group G, exactness of the seβ

α

quence E : 0 → A 󳨀→ B 󳨀→ C → 0 induces exactness of the Hom-Ext sequences E∗

0 → Hom(C, G) → Hom(B, G) → Hom(A, G) 󳨀→ Ext(C, G) β∗

α∗

󳨀→ Ext(B, G) 󳨀→ Ext(A, G) → 0 and E∗

0 → Hom(G, A) → Hom(G, B) → Hom(G, C) 󳨀→ Ext(G, A) β∗

α∗

󳨀→ Ext(G, B) 󳨀→ Ext(G, C) → 0. Proof. By Theorem 1.16.2 and Lemma 1.16.8, it suffices to prove exactness at the groups of extensions. To show that im E ∗ ⊆ ker β∗ , let σ ∈ Hom(A, G). Then β∗ E ∗ (σ) = σEβ = 0 ϕ

ψ

since Eβ splits by Lemma 1.16.8(2). Now let E1 : 0 → G 󳨀→ H 󳨀→ C → 0 ∈ ker β∗ . Then E1 β splits, so by Lemma 1.16.8(2), there is a τ : B → H such that ψτ = β; thus, ψτα = βα = 0. By Lemma 1.2.1 there is a map ρ : A → G with ϕρ = τα; hence, the diagram α

E:

0

→

A ↓ρ

󳨀→

E1 :

0

→

G

󳨀→

ϕ

β

B ↓τ

󳨀→

H

󳨀→

ψ

C ‖

→

0

C

→

0

is commutative. But then E1 = ρE, and we obtain ker β∗ ⊆ imE ∗ . Next we prove exactness at Ext(B, G). Letting E1 ∈ Ext(C, G), we have β∗ (E1 ) ∈ ϕ

ψ

ker α∗ since α∗ β∗ (E1 ) = E1 βα = 0. Now assume that E2 : 0 → G 󳨀→ H 󳨀→ B → 0 ∈ ker α∗ . Since E2 α splits, Lemma 1.16.8(2) shows that there is a τ : A → H such that ψτ = α, and τ is injective since α is. Then βψτ = βα = 0; thus, h + τ(A) 󳨃→ βψ(h) defines a map σ : H/τ(A) → C, and we obtain a commutative diagram

0

→

E2 :

0

→

E3 :

0

→

0 ↓ 0 ↓ G ‖ G ↓ 0

→ ϕ

󳨀→ πϕ

󳨀→

0 ↓ A ↓τ H ↓π H/τ(A) ↓ 0

= ψ

󳨀→ σ

󳨀→

0 ↓ A ↓α B ↓β C ↓ 0

→

0

→

0

→

0.

By Lemma 1.2.4, the row E3 is exact, so we have E2 = E3 β, and therefore, E2 ∈ imβ∗ .

50 | 1 Basic characteristics of abelian groups To show that α∗ is surjective, let E 󸀠 ∈ Ext(A, G). By Theorem 1.5.1, there is an epimorphism ϕ : F → B with F free, and we set K = ker ϕ. Now let L = ϕ−1 (α(A)). Then l 󳨃→ α−1 ϕ(l) defines a surjective homomorphism ϕ󸀠 : L → A with ker ϕ󸀠 = K, so we obtain a commutative diagram ϕ󸀠

E1 :

0

→

K ‖

→

L ↓

󳨀→

E2 :

0

→

K

→

F

󳨀→

ϕ

A ↓α

→

0

B

→

0

with exact rows. Thus, E1 = E2 α. By the first part of this proof, the sequence E1∗

Hom(K, G) 󳨀→ Ext(A, G) → Ext(L, G) is exact. As a subgroup of a free group, L is free; hence, Ext(L, G) = 0, and so we have E 󸀠 = E1∗ (η) for some η ∈ Hom(K, G). Then α∗ (ηE2 ) = ηE2 α = ηE1 = E 󸀠 ; thus, α∗ is surjective. This proves exactness of the first sequence. Next we show exactness at Ext(G, A). The extension αE splits by Lemma 1.16.8(1). Therefore, for any ρ ∈ Hom(G, C), we have α∗ E∗ (ρ) = αEρ = 0. Therefore, im E∗ ⊆ ϕ

ψ

ker α∗ . Now let E1 : 0 → A 󳨀→ H 󳨀→ G → 0 ∈ ker α∗ . Then αE1 splits, so by Lemma 1.16.8(1), there is a τ : H → B such that τϕ = α. This implies that βτϕ = βα = 0; thus, Lemma 1.2.2 yields a map ν : G → C such that νψ = βτ. Hence, we obtain a commutative diagram ϕ

E1 :

0

→

A ‖

󳨀→

E:

0

→

A

󳨀→

α

ψ

H ↓τ

󳨀→

B

󳨀→

β

G ↓ν

→

0

C

→

0.

But then E1 = Eν; thus, E1 ∈ imE∗ as desired. It is clear that im α∗ ⊆ ker β∗ since for any E1 ∈ Ext(G, A), we have β∗ α∗ (E1 ) = ϕ

ψ

βαE1 = 0. Suppose E2 : 0 → B 󳨀→ H 󳨀→ G → 0 ∈ ker β∗ . Then βE2 splits, so Lemma 1.16.8(1) shows that β = τϕ for some τ : H → C which is surjective. Since τϕα = βα = 0, we obtain a map μ = ϕα : A → ker τ. Letting ν : ker τ → H denote the inclusion map, we obtain a commutative diagram 0 ↓ E󸀠 :

0

→

E2 :

0

→

0

→

A ↓α B ↓β C ↓ 0

μ

󳨀→ ϕ

󳨀→ =

0 ↓ ker τ ↓ν H ↓τ C ↓ 0

ψν

󳨀→ ψ

󳨀→ →

0 ↓ G ‖ G ↓ 0 ↓ 0

→

0

→

0

→

0

1.16 Hom and Ext | 51

with exact second row. By Lemma 1.2.4, the row E 󸀠 is exact, and we have αE 󸀠 = E2 . Thus, E2 ∈ imα∗ . ϕ

ψ

To show that β∗ is surjective, let E1 : 0 → C 󳨀→ D 󳨀→ G → 0 ∈ Ext(G, C). Since ϕ

∗

Ext(D, A) 󳨀→ Ext(C, A) → 0 is exact, there is an E 󸀠 ∈ Ext(D, A) such that E 󸀠 ϕ = E; hence, there is a commutative diagram E:

0

→

E󸀠 :

0

→

A ‖

A

α

󳨀→ σ

󳨀→

B ↓ β󸀠

M ↓ ψτ G

β

󳨀→ τ

󳨀→ =

C ↓ϕ

D ↓ψ G

→

0

→

0

with β󸀠 injective (Lemma 1.2.3(2)). To show that the middle column yields an extension Ẽ ∈ Ext(G, B), we need to show that im β󸀠 = ker ψτ. Clearly, im β󸀠 ⊆ ker ψτ since ψτβ󸀠 = ψϕβ = 0. Letting m ∈ ker ψτ, we have τ(m) ∈ ker ψ = im ϕ; thus, τ(m) = ϕβ(b) for some b ∈ B. Then τβ󸀠 (b) = τ(m) shows that β󸀠 (b) − m ∈ ker τ = im σ. Therefore, there is an a ∈ A with σ(a) = β󸀠 (b) − m; hence, m = β󸀠 (b) − β󸀠 α(a) ∈ imβ󸀠 , as desired. It follows that E1 = βE;̃ thus, β∗ is surjective. This completes the proof. In terms of direct sums and direct products, we obtain the following: Theorem 1.16.11. There are natural isomorphisms: Ext(⨁ Ci , A) ≅ ∏ Ext(Ci , A) i∈I

i∈I

and Ext(C, ∏ Aj ) ≅ ∏ Ext(C, Aj ). j∈J

j∈J

Proof. Since every group is an epimorphic image of a free group, we obtain extensions 0 → Bi → Fi → Ci → 0 with Fi free. Recalling that extensions by free groups split, we obtain the exact sequence Hom(Fi , A) → Hom(Bi , A) → Ext(Ci , A) → Ext(Fi , A) = 0. This yields a diagram Hom(⨁ Fi , A) ϕ1 ↓ ∏ Hom(Fi , A)

f1

󳨀→ g1

󳨀→

Hom(⨁ Bi , A) ϕ2 ↓ ∏ Hom(Bi , A)

f2

󳨀→ g2

󳨀→

Ext(⨁ Ci , A) ϕ3 ↓ ∏ Ext(Ci , A)

→

0

→

0

with exact rows and isomorphisms ϕ1 and ϕ2 (Theorem 1.16.3), making the first square commute. We define the map ϕ3 by ϕ3 (f2 (x)) = g2 ϕ2 (x) (x ∈ Hom(⨁ Bi , A)). It is straightforward to verify that ϕ3 is well-defined. Since ϕ3 is a homomorphism, it is an isomorphism by Lemma 1.2.3. The proof of the second assertion is similar.

52 | 1 Basic characteristics of abelian groups We have the following facts: Proposition 1.16.12. Let A and C be groups. Then we have the following: 1. Multiplication by a positive integer n on A or C induces multiplication by n on Ext(C, A); that is, (n1A )∗ = (n1C )∗ = n1Ext(C,A) . 2. If α is an automorphism of A, then α∗ is an automorphism of Ext(C, A). 3. If γ is an automorphism of C, then γ ∗ is an automorphism of Ext(C, A). Proof. To prove (1), let E ∈ Ext(C, A). Since (1A )∗ : E 󳨃→ 1A E = E, we have (n1A )∗ = n(1A )∗ by Lemma 1.16.7; thus, (n1A )∗ : E 󳨃→ nE. Likewise, (n1C )∗ maps E onto nE since E = E1C . (2) is true since αβ = βα = 1A implies α∗ β∗ = β∗ α∗ = 1Ext(C,A) . Likewise, (3) holds since η∗ is the inverse of γ ∗ whenever η is the inverse of γ. As an application of the above proposition we obtain the following: Proposition 1.16.13. Let A, C be groups and n a positive integer. Then the following hold: 1. If nA = 0 or nC = 0, then nExt(C, A) = 0. 2. Ext(ℤ/nℤ, A) ≅ A/nA. 3. If nA = A, then nExt(C, A) = Ext(C, A). Thus, Ext(C, A) is divisible whenever A is. 4. If nA = A and A[n] = 0, then nExt(C, A) = Ext(C, A) and Ext(C, A)[n] = 0. Hence, Ext(C, A) is divisible torsion-free whenever A is. 5. If C[n] = 0, then nExt(C, A) = Ext(C, A). Thus, Ext(C, A) is divisible if C is torsionfree. 6. If nC = 0 and C[n] = 0, then nExt(C, A) = Ext(C, A) and Ext(C, A)[n] = 0. Therefore, Ext(C, A) is divisible torsion-free if C is. Proof. (1): Let E ∈ Ext(C, A). Then (0A )∗ maps E onto E0 (Lemma 1.16.5) and Proposition 1.16.12 shows that (n1A )∗ maps E onto nE. Thus, nE = 0 provided that n1A = 0A . Likewise, we have nE = 0 if n1C = 0C . σ

(2): The extension 0 → ℤ 󳨀→ ℤ → ℤ/nℤ → 0 with σ = n1A induces the exσ

∗

act sequence Hom(ℤ, A) 󳨀→ Hom(ℤ, A) → Ext(ℤ/nℤ, A) → Ext(ℤ, A) = 0. Since σ ∗ : ϕ 󳨃→ nϕ (Proposition 1.16.1) and Hom(ℤ, A) ≅ A, we obtain Ext(ℤ/nℤ, A) ≅ Hom(ℤ, A)/nHom(ℤ, A) ≅ A/nA. β

(3): Since the sequence A 󳨀→ A → 0 with β = n1A is exact, the induced sequence β∗

Ext(C, A) 󳨀→ Ext(C, A) → 0 is exact, and we have β∗ = n1Ext(C,A) . (4) and (6) follow from Proposition 1.16.12(2), (3). α

α∗

(5): Exactness of 0 → C 󳨀→ C with α = n1A induces exactness of Ext(C, A) 󳨀→ Ext(C, A) → 0, where α∗ = n1Ext(C,A) . Exercises. 1. If A = nA for some n ∈ ℕ, then Hom(A, C)[n] = 0 for any group C. 2. If A is any group, then Hom(A, ℚ) ≅ ∏r0 (A) ℚ.

1.17 Tensor products and p-localizations | 53

3. 4. 5. 6. 7. 8.

Show that Hom(A, C) ≅ Hom(A/tA, C) whenever C is torsion-free. If A is torsion, then Hom(A, C) ≅ ∏p∈ℙ Hom(tp A, tp C). If C is any group and n ∈ ℕ, then we have Ext(C, ℤ/nℤ) ≅ Ext(C[n], ℤ/nℤ). Given a group A, we have Ext(C, A) = 0 for all groups C if and only if A is divisible. Given a group C, we have Ext(C, A) = 0 for all groups A if and only if C is free. Let A and C be torsion-free rank 1 groups. Then A ≅ C if and only if Hom(A, C) ≠ 0 and Hom(C, A) ≠ 0.

1.17 Tensor products and p-localizations Many of the problems we solve are best addressed by first localizing to simplify the problem by focusing on a particular prime, and then extending the local result to the more general, global result. Here we construct the tensor product, which will facilitate localization. Let A, C and G be groups. A map b : A × C → G is called bilinear if it satisfies b(a1 + a2 , c) = b(a1 , c) + b(a2 , c) and b(a, c1 + c2 ) = b(a, c1 ) + b(a, c2 ) for all a, a1 , a2 ∈ A and c, c1 , c2 ∈ C. Now let F be the free group with free generators (a, c), where a ∈ A, c ∈ C. Letting Y be the subgroup generated by all elements of the form (a1 + a2 , c) − (a1 , c) − (a2 , c) and (a, c1 + c2 ) − (a, c1 ) − (a, c2 ) with a, a1 , a2 ∈ A and c, c1 , c2 ∈ C, we define the tensor product of A and C to be the group A ⊗ C = F/Y. The coset (a, c) + Y is written a ⊗ c. Then every element of A ⊗ C can be expressed as a (not necessarily unique) finite sum of the form ∑m i=1 ai ⊗ ci (ai ∈ A, ci ∈ C), and we have (a) (a1 + a2 ) ⊗ c = a1 ⊗ c + a2 ⊗ c (b) a ⊗ (c1 + c2 ) = a ⊗ c1 + a ⊗ c2 (c) na ⊗ c = n(a ⊗ c) = a ⊗ nc for all a, a1 , a2 ∈ A, c, c1 , c2 ∈ C and integers n. The following properties are evident for any a ∈ A, c ∈ C and m, n ∈ ℕ: (d) a ⊗ 0 = 0 = 0 ⊗ c. (e) (−a) ⊗ c = −(a ⊗ c) = a ⊗ (−c). (f) If ma = 0 and nc = 0, then gcd(m, n)(a ⊗ c) = 0. (g) If ma = 0 and m|c, then a ⊗ c = 0. (h) If m|a and n|c, then mn|a ⊗ c.

54 | 1 Basic characteristics of abelian groups The tensor map is defined to be t:

A×C (a, c)

→ 󳨃 →

A⊗C a⊗c

and is clearly a bilinear map. We have the following: Theorem 1.17.1. Let A and C be groups and let t : A × C → A ⊗ C be the tensor map. If f : A × C → G is a bilinear map, then there exists a unique homomorphism ϕ : A ⊗ C → G such that the diagram A×C f ↘

t

󳨀→ G

A⊗C ↙ϕ

is commutative. Proof. Letting F be the free group on the set A × C, the correspondence (a, c) 󳨃→ f (a, c) uniquely defines a homomorphism φ : F → G. Let Y be the subgroup of F generated by all elements of the form (a1 + a2 , c) − (a1 , c) − (a2 , c) and (a, c1 + c2 ) − (a, c1 ) − (a, c2 ) with a, a1 , a2 ∈ A and c, c1 , c2 ∈ C. Then φ induces the homomorphism ϕ:

F/Y = A ⊗ C x+Y

→ 󳨃→

G φ(x)

since Y ⊆ ker φ. Letting (a, c) ∈ A × C, we have ϕt(a, c) = ϕ(a ⊗ c) = ϕ((a, c) + Y) = f (a, c); thus, ϕt = f . To prove the uniqueness claim, assume that ψ : A ⊗ C → G is a homomorphism with ψt = f . Then ψ(a ⊗ c) = ψt(a, c) = f (a, c); hence, ψ = ϕ. The next result contains some basic properties of tensor products: Proposition 1.17.2. Let A and C be groups. Then we have the following: 1. A ⊗ C ≅ C ⊗ A. 2. ℤ ⊗ C ≅ C. 3. If A or C is a p-group (torsion), then A ⊗ C is a p-group (torsion). 4. If A is a p-group and C is p-divisible, then A ⊗ C = 0. 5. If A or C is p-divisible, then A ⊗ C is p-divisible. Proof. To prove (1), let F be the free group on the set A × C. Then (a, c) 󳨃→ c ⊗ a defines an epimorphism μ : F → C ⊗ A with F/ ker μ = A ⊗ C. To show (2), consider the map σ : C → ℤ⊗C given by c 󳨃→ 1⊗c. Then σ is surjective since elements of ℤ ⊗ C have the form ∑ni=1 (ri ⊗ ci ) = ∑ni=1 (1 ⊗ ri ci ) = 1 ⊗ (∑ni=1 ri ci ) with ri ∈ ℤ and ci ∈ C (i = 1, . . . , n). The map f : ℤ × C → C given by (m, c) 󳨃→ mc is bilinear, so by Theorem 1.17.1 there is a homomorphism ϕ making the diagram ℤ×C f ↘

t

󳨀→ C

ℤ⊗C ↙ϕ

1.17 Tensor products and p-localizations | 55

commute. Then ϕ : 1 ⊗ c 󳨃→ f (1, c) = c shows that ϕ and σ are inverse to each other; thus, σ is an isomorphism. For (3), use (c) and (d), whereas (4) and (5) follow from (g) and (h), respectively. Tensor products commute with direct sums: Theorem 1.17.3. If A = ⨁i∈I Ai and C = ⨁j∈J Cj , then A ⊗ C ≅ ⨁i,j (Ai ⊗ Cj ). Proof. We claim that by Proposition 1.17.2(1), it suffices to prove that A ⊗ C ≅ ⨁i∈I (Ai ⊗ C). Assume that we know A ⊗ C ≅ ⨁i (Ai ⊗ C). Since Ai ⊗ C ≅ C ⊗ Ai , we have A ⊗ C ≅ ⨁i (Ai ⊗ C) ≅ ⨁i (C ⊗ Ai ) ≅ ⨁i (⨁j (Cj ⊗ Ai )). The last isomorphism holds because for fixed i, C ⊗ Ai ≅ ⨁j (Cj ⊗ Ai ) by our assumption. Therefore, A ⊗ C ≅ ⨁i,j (Ai ⊗ Cj ) follows, proving the claim. For i ∈ I let σi : Ai → A be the inclusion and πi : A → Ai the projection. Since f :

A×C (a, c)

B = ⨁i∈I (Ai ⊗ C) ∑(πi (a) ⊗ c)

→ 󳨃→

is a bilinear map, there is a homomorphism ϕ : A ⊗ C → B such that f = ϕt, where t : A × C → A ⊗ C is the tensor map. Clearly, ϕ is surjective since the elements πi (a) ⊗ c generate B. For each i ∈ I, there is a homomorphism ηi : Ai ⊗ C → A ⊗ C, making the diagram ti

Ai × C

󳨀→

fi ↘

A⊗C

Ai ⊗ C ↙ηi

commute, where ti is the tensor map and fi : (ai , c) 󳨃→ σi (ai ) ⊗ c; thus, ηi : ai ⊗ c 󳨃→ σi (ai ) ⊗ c. This yields a homomorphism η:

B ∑ xi

→ 󳨃 →

A⊗C ∑ ηi (xi )

with xi ∈ Ai ⊗ C. But then ηf (a, c) = η(∑(πi (a) ⊗ c)) = ∑ ηi (πi (a) ⊗ c) = ∑(σi πi (a) ⊗ c) = a ⊗ c; hence, ηϕ(a ⊗ c) = ηϕt(a, c) = ηf (a, c) = a ⊗ c. This shows that ϕ is injective; thus, ϕ is an isomorphism. Given homomorphisms α : A → A󸀠 and γ : C → C 󸀠 , we obtain a bilinear map f : A × C → A󸀠 ⊗ C 󸀠 defined by (a, c) 󳨃→ α(a) ⊗ γ(c). By Theorem 1.17.1, we obtain a homomorphism ϕ : A ⊗ C → A󸀠 ⊗ C 󸀠 such that the diagram A×C f ↘

t

󳨀→ A󸀠 ⊗ C 󸀠

A⊗C ↙ϕ

56 | 1 Basic characteristics of abelian groups is commutative. Writing α ⊗ γ instead of ϕ, we have α ⊗ γ : a ⊗ c 󳨃→ α(a) ⊗ γ(a). Therefore, given a group G, the map α : A → A󸀠 induces the homomorphism α∗ = α ⊗ 1G : A ⊗ G → A󸀠 ⊗ G. Tensoring with G preserves right-exactness of short exact sequences: Theorem 1.17.4 (Cartan–Eilenberg [12]). Given any group G, exactness of the sequence α

β

0 → A 󳨀→ B 󳨀→ C → 0 induces exactness of the sequence β∗

α∗

A ⊗ G 󳨀→ B ⊗ G 󳨀→ C ⊗ G → 0. Proof. Let a ∈ A and g ∈ G. Then β∗ α∗ (a ⊗ g) = β∗ (α(a) ⊗ g) = βα(a) ⊗ g = 0; thus, im α∗ ⊆ ker β∗ . Therefore, β∗ induces the homomorphism η : K = (B ⊗ G)/imα∗ → C ⊗ G with η : (b ⊗ g) + im α∗ 󳨃→ β∗ (b ⊗ g). To prove that ker β∗ ⊆ imα∗ , we show that η is injective. For every c ∈ C, there is a b ∈ B such that β(b) = c and if β(b) = β(b󸀠 ) for some b󸀠 ∈ B, then (b ⊗ g) − (b󸀠 ⊗ g) = (b − b󸀠 ) ⊗ g = α(a) ⊗ g for some a ∈ A. Thus, the map f :

C×G (c, g)

→ 󳨃→

K (b ⊗ g) + im α∗

is well-defined. Then f is a bilinear map, so there is a homomorphism ϕ : C ⊗ G → K with ϕ(c ⊗ g) = (b ⊗ g) + im α∗ (Theorem 1.17.1). Then ϕη = 1K ; thus, η is injective. It is clear that β∗ is surjective since C ⊗ G is generated by elements of the form β(b) ⊗ g. Of particular interest is tensoring of G with ℤp (= the p-localization of ℤ at the prime p), resulting in Gp = G ⊗ ℤ p , which is called the p-localization of G. An examination of these groups is in order since they form the basis for many of the classification results: Lemma 1.17.5. Let G be a group and p a prime. Then we have the following: 1. Every element of Gp can be written as x ⊗ n1 for some x ∈ G and n ∈ ℤ with p ∤ n. 2. Gp is a p-local group, that is, a ℤp -module by defining r(x ⊗ z) = x ⊗ rz for r ∈ ℤp . Thus, every element of Gp has a unique q-divisor for any prime q ≠ p.

1.17 Tensor products and p-localizations | 57

3.

Multiplication by a prime ≠ p in Gp is an automorphism. Hence, Gp has no q-torsion for any prime q ≠ p.

Proof. An arbitrary element of Gp is of the form x1 ⊗

m1 n1

+ ⋅ ⋅ ⋅ + xk ⊗

and mi , ni ∈ ℤ with p ∤ ni for i = 1, . . . , k. This reduces to n=

∏kj=1 nj .

mk , nk

[∑ki=1 mi (∏j=i̸

where xi ∈ G

nj )xi ] ⊗

1 n

with

To prove (2), we fix r ∈ ℤp and consider the bilinear map f : G × ℤp → Gp given by (x, z) 󳨃→ x⊗rz. By Theorem 1.17.1, we obtain a homomorphism ϕ : Gp → Gp which maps x ⊗ z onto x ⊗ rz. Thus, for any given x ∈ G and r, z ∈ ℤp , the product r(x ⊗ z) = x ⊗ rz is well-defined. This yields a ℤp -module structure of Gp . (3) is a key characteristic of p-local groups, and we include a proof for the sake of completeness. Let q be a prime ≠ p. The map σ : Gp → Gp given by y 󳨃→ qy is clearly surjective, and since σ is a ℤp -module homomorphism, K = ker σ is a submodule of Gp . If y ∈ K, then y󸀠 = q1 y ∈ K, and we have y = qy󸀠 = 0; hence, K = 0. Therefore, σ is an automorphism. This lemma leads us to multiple ways of characterizing p-local groups: Theorem 1.17.6. Let A be a group. Then the following are equivalent: 1. A is a ℤp -module; 2. A ≅ A ⊗ ℤp ; 3. A ≅ C ⊗ ℤp for some group C; 4. For every q ≠ p, A is uniquely q-divisible. Proof. Suppose A is a ℤp -module. Define f : A × ℤp → A by f (a, r) = ra for all a ∈ A and r ∈ ℤp . Then Theorem 1.17.1 defines a homomorphism ϕ : A ⊗ ℤp → A, which is clearly surjective. Let a ⊗ n1 ∈ ker ϕ for some a ∈ A and n not divisible by p. Then 0 = f (a, n1 ) = n1 a, and hence, a = 0 and a ⊗ n1 = 0. Thus, ϕ is an isomorphism, proving (1) implies (2). Clearly, (2) implies (3), and (3) implies (4) by Lemma 1.17.5(2). Now suppose (4). Let a ∈ A and m/n ∈ ℤp . Since n is relatively prime to p, there is a unique b ∈ A such that a = nb. Define mn a = mb. It is straightforward to verify that this defines a ℤp -module, proving (4) implies (1). Note that (4) implies an important property of p-local groups: They have no q-torsion for any prime q ≠ p. As we will see later, heights and torsion elements are key characteristics for classifying groups. By converting G to Gp , we remove all torsion—except for p-torsion—and make all q-heights infinite for all primes q ≠ p. We will soon see that localization preserves p-heights, exact sequences, and Ulm invariants. The natural map νp : is called the localization map.

G x

→ 󳨃 →

Gp = G ⊗ ℤ p x⊗1

58 | 1 Basic characteristics of abelian groups Now we take a closer look at the mapping. First we note that for any x ⊗ n1 ∈ Gp , we have n(x ⊗ n1 ) ∈ νp (G). On the other hand, suppose pk (x ⊗ n1 ) = y ⊗ 1 ∈ νp (G). Then since n and pk are relatively prime, for some integers a and b, x⊗ n1 = a(x⊗1)+b(y⊗1) ∈ νp (G). This gives us the following characteristics of νp (G): (i) If pk z ∈ νp (G) for some z ∈ Gp , then z ∈ νp (G). (j) Every element of Gp has a multiple in νp (G). The following result will be needed: Lemma 1.17.7. Let f : G → H be a homomorphism and p a prime such that ker f is p-divisible and coker f has no p-torsion. Then we have the following: 1. If α is an ordinal, then pα H/f (pα G) has no p-torsion. 2. f is p-height-preserving. Proof. The first assertion is proved using transfinite induction. For α = 0, there is nothing to do, so let α > 0 and assume that the claim holds for all ordinals < α. Let y ∈ pα H such that pn y ∈ f (pα G) for some positive integer n. We need to show that y ∈ f (pα G). If α − 1 exists, we have y = pz for some z ∈ pα−1 H. Then pn+1 z = pn y ∈ f (pα G) ⊆ f (pα−1 G); hence, z ∈ f (pα−1 G) by the induction hypothesis. It follows that z = f (x) for some x ∈ pα−1 G; hence, y = pz = pf (x) = f (px) ∈ f (pα G), as desired. If α is a limit ordinal, then pn y ∈ f (pα G) = f (⋂β 0, and assume that the inclusion holds for all ordinals β < α. Suppose that x + A + B/A ∈ pα ((G/A)/(B/A)). If α − 1 exists, then there is a y + A + B/A ∈ pα−1 ((G/A)/(B/A)) such that py + A + B/A = x + A + B/A.

(∗)

By the induction hypothesis, we have y + A ∈ pα−1 (G/A) + B/A; consequently, x + A ∈ pα (G/A) + B/A. Finally, assume that α is a limit ordinal, and let β < α. Then there is a y + A + B/A ∈ pβ ((G/A)/(B/A)) satisfying (∗). By the induction hypothesis, we obtain y + A ∈ pβ (G/A) + B/A; thus, there is a b ∈ B such that x + b + A ∈ pβ (G/A). But then x + B ∈ pβ (G/B), which shows that x + B ∈ pα (G/B). Since B is p-nice in G, there is a b󸀠 ∈ B such that x + b󸀠 ∈ pα G. Thus, x + b󸀠 + A ∈ pα (G/A), and again, it follows that x + A ∈ pα (G/A) + B/A.

2.2 Families of subgroups and the third axiom of countability |

63

To show the second assertion, let x ∈ G \B, and use Proposition 2.1.1(2) repeatedly. Then |x + A + B/A|(G/A)/(B/A) = |x + b + A|G/A for some b ∈ B and |x + b + A|G/A = |x + b + a|Gp p p p

for some a ∈ A. This yields |x + b + a|Gp = |x + B|G/B p ; therefore, B is p-nice in G.

Niceness is not transitive (see, for example, Fuchs [29] Chapter 11, Section 2, Exercise 7), but we have the following: Lemma 2.1.5. Let G be an R-module, p ∈ R a prime, and suppose that A, B and C are submodules of G such that A ⊆ B ⊆ C. If every coset b + A (b ∈ B) has an element of maximal p-height and every coset c + B (c ∈ C) has an element of maximal p-height, then every coset z + A (z ∈ C) has an element of maximal p-height. Here all p-heights are computed in the module G. Proof. Let z ∈ C. Then the coset z + A has an element of maximal p-height if z ∈ B, so let us assume that z ∈ ̸ B. By assumption, there is a b ∈ B such that |z + b| ≥ |z + x| for all x ∈ B. If b ∈ A, the assertion follows. If b ∈ ̸ A, there is an a ∈ A such that |b + a| ≥ |b + y| for all y ∈ A. Let c ∈ A. Then |z − a| = |b + a − (z + b)| ≥ |b + a| ∧ |z + b| ≥ |b − c| ∧ |z + c| = |z + b − (z + c)| ∧ |z + c| ≥ |z + b| ∧ |z + c| = |z + c|. Consequently, z − a is an element of maximal p-height in the coset z + A. Lemma 2.1.6. Let G be an R-module, p ∈ R a prime, and β an ordinal. If A is a submodule of pβ G, then A is p-nice in pβ G if and only if A is p-nice in G. Proof. Suppose A is p-nice in pβ G. Let x ∈ G. If x ∈ ̸ pβ G, then for all a ∈ A |x + a|G = |x|G ∧ |a|G = |x|G , proving x proper with respect to A. So suppose x ∈ pβ G. Since A is β β p-nice in pβ G, there is some a ∈ A such that |x + a|p G ≥ |x + a󸀠 |p G for all a󸀠 ∈ A. Since β β x + a and x + a󸀠 ∈ pβ G, we have |x + a|G = β + |x + a|p G ≥ β + |x + a󸀠 |p G = |x + a󸀠 |G by Lemma 1.10.4(2). Now suppose A is p-nice in G. Let x ∈ pβ G. Then there is a ∈ A with β β |x + a|G ≥ |x + a󸀠 |G for all a󸀠 ∈ A. Then by Lemma 1.10.4(2), β + |x + a|p G ≥ β + |x + a󸀠 |p G ; β β so by left cancellation of ordinals, |x + a|p G ≥ |x + a󸀠 |p G for all a󸀠 ∈ A. Exercises. 1. If A is a nice subgroup of a reduced p-group G, then A = G whenever G/A is divisible. 2. A pure subgroup of a p-group needs not be nice. 3. (Hill [47]) Let A be a submodule of an R-module G, p ∈ R a prime, and β an ordinal. (a) If A is p-nice in G, then A ∩ pβ G is p-nice in pβ G. (b) Let A and G be as in Proposition 2.1.3. If A is p-nice in G and A ∩ pβ G is p-nice in pβ G, then A is p-nice in G.

2.2 Families of subgroups and the third axiom of countability In [47], Hill introduced the following axioms of countability for p-groups:

64 | 2 Simply presented groups Axiom 1. A group is countable. Axiom 2. A group is a direct sum of countable groups. Axiom 3. A group G has a system 𝒞 of nice subgroups satisfying the following conditions: (H1) 0 ∈ 𝒞 ; (H2) if Ni ∈ 𝒞 (i ∈ I), then ∑i∈I Ni ∈ 𝒞 ; (H3) if N ∈ 𝒞 and S is any countable subset of G, then there is an M ∈ 𝒞 such that ⟨S, N⟩ ⊆ M and |M/N| ≤ ℵ0 . Clearly we have the following: Axiom 1 ⇒ Axiom 2 ⇒ Axiom 3. Axiom 3 is also called the third axiom of countability or Hill’s condition. Given any R-module G, a system 𝒞 of submodules of G satisfying (H1)–(H3) is called an H(ℵ0 )-family in G (see Fuchs–Hill [30], in which H(κ)-families were introduced for arbitrary infinite cardinals κ). In the case where G is a p-group, then an H(ℵ0 )-family of nice subgroups is simply called a nice system. For instance, a divisible p-group G has a nice system since every subgroup of G is nice. In [38], Griffith replaced (H2) by the weaker condition (G2) 𝒞 is closed under ascending unions. A system 𝒞 of subgroups of a group G is called a G(ℵ0 )-family in G if it satisfies (H1), (G2), and (H3). Obviously, every H(ℵ0 )-family in G is a G(ℵ0 )-family. A wellordered ascending chain of subgroups N0 ⊆ N1 ⊆ ⋅ ⋅ ⋅ ⊆ Nα ⊆ ⋅ ⋅ ⋅ (α < λ) is called smooth if for every limit ordinal α < λ we have Nα = ⋃β β+1, contradicting our choice of x. Therefore, y is proper with respect to B, and we have y ∉ B since |y| ≠ ∞. By Lemma 1.12.4, ϕ extends to a p-height-preserving isomorphism ϕ󸀠 : A1 = ⟨A, x⟩ → B1 = ⟨B, y⟩ by sending x onto y. To show that A1 (β) = A(β), let a + rx + g ∈ A1 (β), where a ∈ A, r ∈ R and g ∈ pβ+1 G. If p|r, then a + rx + g ∈ A(β), and if p ∤ r, then there are s, t ∈ R such that sp + tr = 1. Thus, |a + rx| = |ta + trx| = |ta − spx + x| ≤ |x| = β. Letting a󸀠 = ta − spx, we have a󸀠 + x = t(a + rx), and so |a󸀠 + x| = β since |a + rx + g| ≥ β. Then a󸀠 + x is proper with respect to A, and |p(a󸀠 + x)| = |pg| > β + 1, again contradicting our choice of x. Therefore, we have A1 (β) = A(β). Likewise, we obtain B1 (β) = B(β). In either case, we constructed ϕ󸀠 : A1 → B1 —as required—such that fβ maps A1 (β)/A(β) onto B1 (β)/B(β). Now assume that α ≠ β is an ordinal. Notice that A1 and B1 are p-nice since finite extensions of p-nice submodules are p-nice. Then if α < β, we have x ∈ pα+1 G; hence, A1 (α) = A(α) and likewise, B1 (α) = B(α). If α > β, let a + rx + g ∈ A1 (α), where a ∈ A, r ∈ R and g ∈ pα+1 G. If p|r, then a + rx + g ∈ A(α). If p ∤ r, then there are s, t ∈ R such that sp + tr = 1. Hence, |a + rx| = |ta + trx| = |ta − spx + x| ≤ |x| = β < α, therefore, |a + rx + g| < α, a contradiction. Thus, it follows that A1 (α) = A(α). Likewise, we have B1 (α) = B(α). Now consider the case α = ∞, and let a ∈ A and r ∈ R such that a + rx ∈ A1 (∞). If p|r, then a + rx ∈ A(∞), and if p ∤ r, then again, there are s, t ∈ R such that sp + tr = 1, and we have |a + rx| = |ta + trx| = |ta − spx + x| ≤ |x| = β < ∞, a contradiction. Therefore, A1 (∞) = A(∞), and by symmetry, B1 (∞) = B(∞). Thus, (2) is true whenever |x| ≠ ∞. Finally, suppose that |x| = ∞. First we assume that A1 (∞) = A(∞). Let ϕ󸀠 : A1 = ⟨A, x⟩ → B1 = ⟨B, y⟩ be any p-height-preserving isomorphism extending ϕ by sending x onto some y ∈ H (see Lemma 1.14.1). To show that B1 (∞) = B(∞), let b + ry ∈ B1 (∞) with b ∈ B and r ∈ R. Then ϕ−1 (b) + rx ∈ A1 (∞) = A(∞); thus p|r, and we obtain B1 (∞) = B(∞). Now assume that A1 (∞) ≠ A(∞). We wish to find an element x󸀠 ∈ x + A such that 󸀠 x ∈ (p∞ G)[p]. There exists a z ∈ A1 (∞) \ A(∞), say z = rx + a ∈ (p∞ G)[p] with r ∈ R (p ∤ r) and a ∈ A. Then 1 = sr + tp for some s, t ∈ R; thus, sz = srx + sa = (1 − tp)x + sa =

70 | 2 Simply presented groups x + a󸀠 with a󸀠 = sa − tpx ∈ A. Consequently, x󸀠 = x + a󸀠 is an element of (p∞ G)[p], as desired. Let y ∈ (p∞ H)[p] such that f∞ (x󸀠 + A(∞)) = y + B(∞). By Lemma 1.12.4, ϕ extends to a p-height-preserving isomorphism ϕ󸀠 : A1 = ⟨A, x󸀠 ⟩ → B1 = ⟨B, y⟩ by sending x󸀠 onto y. We have A1 (∞) = ⟨x 󸀠 , A(∞)⟩, and B1 (∞) = ⟨y, B(∞)⟩; thus, f∞ maps A1 (∞)/A(∞) onto B1 (∞)/B(∞), as required. Since |x| = ∞ = |y|, it is clear that, in any case, we have A1 (α) = A(α) and B1 (α) = B(α) for each ordinal α. Therefore, (2) is true if |x| = ∞. (3) follows from (2), so the proof is complete. Now we can prove the following result: Theorem 2.3.2 (Hill [47], Walker [148], Warfield [154]). Let G and H be R-modules, where R = ℤ or R = ℤp for some prime p. Suppose that A and B are p-nice submodules of G and H, respectively, such that G/A and H/B are p-groups with nice systems and uα (G, A) = uα (H, B) for every α ∈ Ord∞ . Then any p-height-preserving isomorphism ϕ : A → B can be extended to an isomorphism G → H. Proof. Let 𝒞 and 𝒟 be nice systems for G/A and H/B, respectively. For any α, an ordinal or ∞, let fα : (pα G)[p]/A(α) → (pα H)[p]/B(α) be an isomorphism, and define ℱ to be the system of all p-height-preserving isomorphisms C → D extending ϕ satisfying the following properties: 1. C/A ∈ 𝒞 and D/B ∈ 𝒟; 2. fα maps C(α)/A(α) onto D(α)/B(α) for all α. The system ℱ is partially ordered in an obvious manner and is nonempty since ϕ ∈ ℱ . Thus, ℱ possesses a maximal element ϕ0 : C → D by Zorn’s lemma. We will show that C = G and D = H. Assume that there is an x ∈ G \ C. By (2), we have uα (G, C) = uα (H, D) for every α, an ordinal or ∞; hence, fα induces an isomorphism gα : (pα G)[p]/C(α) → (pα H)[p]/D(α). By Lemma 2.1.4(2), the modules C and D are p-nice. Then Lemma 2.3.1 yields a p-height-preserving isomorphism ϕ1 : A1 = ⟨C, x⟩ → B1 such that A1 /C and B1 /D are finite and gα maps A1 (α)/C(α) onto B1 (α)/D(α) for all α. Consequently, each map fα maps A1 (α)/A(α) onto B1 (α)/B(α). Since C/A ∈ 𝒞 and ⟨A, x⟩/A is finite, there is a C1 /A ∈ 𝒞 such that A1 /A ⊆ C1 /A and |C1 /A| ≤ ℵ0 . We write C1 = ⟨A1 , x11 , x12 , . . . ⟩. Using Lemma 2.3.1 again, we extend ϕ1 to a p-height-preserving isomorphism ϕ󸀠1 : A󸀠1 = ⟨A1 , x11 ⟩ → B2 = ⟨B1 , y⟩ such that fα maps A󸀠1 (α)/A(α) onto B2 (α)/B(α) for each α. Since D/B ∈ 𝒟 and ⟨B, y⟩/B is finite, there is a D2 /B ∈ 𝒟 such that B2 /B ⊆ D2 /B and |D2 /B2 | ≤ ℵ0 , and we write D2 = ⟨B2 , y21 , y22 , . . . ⟩. Again, we apply Lemma 2.3.1 and extend ϕ󸀠1 to a p-height-preserving isomorphism ϕ2 : A2 = ⟨A󸀠1 , z⟩ → B󸀠2 = ⟨B2 , y21 ⟩

2.3 Classification of p-groups with nice systems | 71

so that fα maps A2 (α)/A(α) onto B󸀠2 (α)/B(α) for each α. Then there is a C2 /A ∈ 𝒞 such that A2 /A ⊆ C2 /A and |C2 /A2 | ≤ ℵ0 , and we write C2 = ⟨A2 , x21 , x22 , . . . ⟩. The map ϕ2 extends to ϕ󸀠2 : A󸀠2 = ⟨A2 , x12 , x21 ⟩ → B3 such that fα maps A󸀠2 (α)/A(α) onto B3 (α)/B(α) for each α. There is a D3 /B ∈ 𝒟 such that B3 /B ⊆ D3 /B and |D3 /B3 | ≤ ℵ0 , so we write D3 = ⟨B3 , y31 , y32 , . . . ⟩ and extend ϕ󸀠2 to ϕ3 : A3 → B󸀠3 = ⟨B3 , y22 , y31 ⟩. Continuing in this way, with each A󸀠n adding new elements from each Ci with 1 ≤ i ≤ n, and likewise for B󸀠n ; this yields sequences C ⊆ A1 ⊆ A󸀠1 ⊆ A2 ⊆ A󸀠2 ⊆ ⋅ ⋅ ⋅ and D ⊆ B1 ⊆ B󸀠1 ⊆ B2 ⊆ B󸀠2 ⊆ ⋅ ⋅ ⋅ , and a p-height-preserving isomorphism ∞

∞

∞

∞

i=1

i=1

i=1

i=1

ϕ̃ : C̃ = ⋃ Ai = ⋃ Ci → D̃ = ⋃ Bi = ⋃ Di ̃ ∈ 𝒞 and D/B ̃ ∈ 𝒟. Moreover, extending ϕ. Since 𝒞 and 𝒟 are nice systems, we have C/A ̃ ̃ fα induces an isomorphism C(α)/A(α) → D(α)/B(α) for all α, and we conclude that ϕ̃ ∈ ℱ . Since x ∈ C,̃ this contradicts the maximality of ϕ; hence, C = G. By symmetry, we have D = H, as desired. As a corollary, we obtain Hill’s generalization of Ulm’s theorem, the classification of p-groups having nice systems in terms of Ulm invariants: Theorem 2.3.3 (Hill [47]). Let G and H be p-groups having nice systems. Then G and H are isomorphic if and only if u(α, G) = u(α, H) for each α ∈ Ord∞ . Another application of Theorem 2.3.2 is the following extension theorem, which will be needed in Section 2.9: Theorem 2.3.4 (Warfield [154]). Let G and H be ℤp -modules and A a p-nice submodule of G such that G/A is a p-group with a nice system. If ϕ : A → H is a homomorphism which does not decrease p-heights, then ϕ extends to a homomorphism G → H. G G H G H Proof. If a ∈ A, then |ϕ(a)|H p ≥ |a|p yields |a|p ∧ |ϕ(a) + h|p = |a|p ∧ |h|p for all h ∈ H; thus, the map

φ:

A⊕H (a, h)

→ 󳨃→

A⊕H (a, ϕ(a) + h)

is a p-height-preserving isomorphism. By Theorem 2.3.2, φ extends to an automorphism φ󸀠 of G ⊕ H. Let σ : G → G ⊕ H be the natural inclusion and μ : G ⊕ H → H the natural projection. Then μφ󸀠 σ : G → H extends ϕ, as desired.

72 | 2 Simply presented groups Exercises. 1. Find a p-group which does not have a nice system. [Hint: Section 1.14, Exercise 2.] 2. (Fuchs [27]) Let G and H be reduced p-groups, and assume that ϕ : A → B is a height-preserving isomorphism, where A is a nice subgroup of G and B is any subgroup of H. If G/A has a nice composition series such that uα (G, A) ≤ uα (H, B) for all ordinals α, then ϕ extends to a height-preserving isomorphism G → H 󸀠 for some subgroup H 󸀠 of H. 3. (Fuchs [27]) Let G be a reduced p-group with nice composition series, and let H be any p-group. Then G is isomorphic to an isotype subgroup of H if and only if u(α, G) ≤ u(α, H) for all ordinals α. [Hint: Exercise 2.] 4. (Fuchs [27]) Let G and H be p-groups and ϕ : A → H a homomorphism, where A is a nice subgroup of G. If G/A has a nice composition series and if ϕ does not decrease heights, then ϕ extends to a homomorphism G → H which does not decrease heights either. [Hint: Exercise 2.]

2.4 Totally projective p-groups and generalized Prüfer groups This section presents another way of looking at the groups we just classified, using a homological approach introduced by Nunke [124]. A p-group G is called totally projective if it satisfies pα Ext(G/pα G, H) = 0 for all ordinals α and groups H. A torsion group is said to be totally projective if each of its p-torsion parts is totally projective. The following properties are immediate: (a) Divisible p-groups are totally projective. (b) Cyclic p-groups are totally projective. (c) The class of totally projective p-groups is closed under arbitrary direct sums and direct summands. (d) If G is a totally projective p-group, then so is G/pσ G for all ordinals σ. Following Nunke [124], we will construct a reduced p-group Hα for every ordinal α such that (i) Hα has length α; (ii) pα Hα+1 is cyclic of order p and Hα+1 /pα Hα+1 ≅ Hα ; (iii) Hα = ⨁β |(β, α1 , . . . , αn , x)+A| ≥ β. Since β was arbitrary, it follows that |(α1 , . . . , αn , x) + A| ≥ α1 , completing the induction. Every element of B ⊕ G may be written in the form a = a1 (α11 , . . . , α1n1 , x1 ) + ⋅ ⋅ ⋅ + ak (αk1 , . . . , αknk , xk ) + xk+1 , where x1 , . . . , xk+1 ∈ G, the (αi1 , . . . , αini , xi ) are distinct, and a1 , . . . , ak ∈ ℤp \ {0}. Since p-heights are unaffected by multiplication by integers relatively prime to p, we may assume a1 , . . . , ak ∈ ℤ. If 0 < ai < p for all i = 1, . . . , k, we say a is in canonical form. We define h(a) = min{α11 , . . . , αk1 , v(xk+1 )}. We prove by induction on m, the maximum number of ordinals in the terms, that there is an a󸀠 ∈ a + A in the canonical form and h(a󸀠 ) ≥ h(a). If m = 0, we have an element of G which is immediately in the form. Suppose it holds for m − 1, and write ai = a󸀠i + pbi , where 0 ≤ a󸀠i < p for i = 1, . . . , n. Let a󸀠 = a󸀠1 (α11 , . . . , α1n1 , x1 ) + ⋅ ⋅ ⋅ + a󸀠k (αk1 , . . . , αknk , xk ) + b1 (α12 , . . . , α1n1 , x1 ) + ⋅ ⋅ ⋅ + bk (αk2 , . . . , αknk , xk ) + xk+1 . Then a + A = a󸀠 + A, and each term of a󸀠 is either already in the canonical form, or can be put in the canonical form by the induction hypothesis. Gather terms and repeat as necessary to put a󸀠 in canonical form. We also note that h(a󸀠 ) ≥ h(a), completing the induction. The same argument shows that h(a󸀠 ) ≥ h(b) for all b ∈ a + A. Now we prove that the canonical form is unique. Suppose, on the contrary, (a1 − a󸀠1 )(α11 , . . . , α1n1 , x1 ) + ⋅ ⋅ ⋅ + (ak − a󸀠k )(αk1 , . . . , αknk , xk ) + xk+1 ∈ A, where 0 ≤ ai < p and 0 ≤ a󸀠i < p for i = 1, . . . , k. But this has no nonzero coefficients divisible by p, whereas every element of A does. Suppose a is in canonical form. From the claim |a + A| ≥ h(a). We will prove by induction on h(a) that |a+A| = h(a). Suppose a+A = pb+A for some b ∈ B⊕G. The case of h(a) = ∞ follows immediately from the claim, so we may assume h(a) ≠ ∞. Write b + A = b1 (β11 , . . . , β1m1 , y1 ) + ⋅ ⋅ ⋅ + bk (βk1 , . . . , βkmk , yk ) + yk+1 + A, where 0 < bi < p for i = 1, . . . , k. Then pb + A = a󸀠 + A, where a󸀠 = b1 (β12 , . . . , β1m1 , y1 ) + ⋅ ⋅ ⋅ + bk (βk2 , . . . , βkmk , yk ) + pyk+1 + A. Note that h(a󸀠 ) > h(b). Let a󸀠󸀠 ∈ a󸀠 + A be in canonical form. Then, h(a󸀠󸀠 ) ≥ h(a󸀠 ). Now a󸀠󸀠 + A = a󸀠 + A = pb + A = a + A. By the uniqueness of the canonical form, we have h(a) = h(a󸀠󸀠 ) ≥ h(a󸀠 ) > h(b). By the induction hypothesis, |b+A| = h(b) < h(a). This completes the induction. As a special case, we find that for all x ∈ S, |x󸀠 | = v(x) with x󸀠 = x + A, the image of x in G󸀠 , as desired. If we take S = G we get the usual form of the theorem: Theorem 2.7.3 (Richman–Walker [134]). Let G be a ℤp -module and v a p-valuation on G. Then there is a ℤp -module H with submodule G󸀠 ≅ G, where H is torsion over G󸀠 , H/G󸀠 is simply presented, and for every x ∈ G and its image x󸀠 ∈ G󸀠 , |x 󸀠 |H p = v(x). The group H in Theorem 2.7.3 is called a Richman–Walker group. Recall that this was introduced in Section 1.11, without proof, as an example of a group with any desired height structure.

2.7 Richman–Walker groups and variations | 81

Theorem 2.7.4 (Walker [148]). Let p be a prime. If α is an ordinal, then there exists a simply presented p-group Pα such that pα Pα is cyclic of order p. Proof. Given α, let G = ⟨x⟩ be generated by some x of order p, and define a valuation on G by v(ax) = α for all a not divisible by p. Let S = {x}. Construct H as in Theorem 2.7.2, and call it Pα . Then the construction shows that |ax| = α for all a relatively prime to p and |y| < α for all y ∈ Pα \ ⟨x⟩. Thus, pα Pα = ⟨x⟩, cyclic of order p. By Proposition 2.5.5, the group Pα is simply presented. Let us take a closer look at this important group Pα , called a Walker group. Since x is an element of height α, we may replace x by α in the representation, obtaining the simple presentation Pα = ⟨X; Λ⟩, where X is the set of all finite sequences of ordinals (α1 , . . . , αn ) with α = αn > αn−1 > ⋅ ⋅ ⋅ > α1 , and Λ is the set of all relations of the forms p(α1 , . . . , αn ) = (α2 , . . . , αn )

(n > 1)

and p(α) = 0. Note that up (β, Pα ) = 1 if β = α and 0 otherwise. Corollary 2.7.5 (Warfield [152, 154]). Let p be a prime and α an Ulm sequence. Then there is a simply presented ℤp -module Hx of torsion-free rank 1 containing an element of infinite order with Ulm sequence α. Proof. Let G = ⟨x⟩ for some x of infinite order, and write α = (αk ). Define a valuation on ⟨x⟩ by v(pk x) = αk , and construct Hx as in Theorem 2.7.3. By construction, Hx is simply presented. Since pn (α1 , . . . , αn , x) ∈ ⟨x⟩, Hx /⟨x⟩ is torsion, and so, since x has infinite order, Hx has torsion-free rank 1. Also, |pk x| = v(pk x) = αk , so the Ulm sequence of x is α. We may extend this result to general groups: Theorem 2.7.6. Let G be a group and vp a p-valuation on G for each prime p. Then there is a group H such that G is a subgroup of H, H/G is torsion, H/G is simply presented, and for all x ∈ G and p prime, |x|H p = vp (x). Proof. For each prime p, let B(p) be the free group generated by all finite sequences (αp1 , . . . , αpn , x) with 0 ≠ x ∈ G and αp1 < ⋅ ⋅ ⋅ < αpn < vp (x) ordinals or ∞ such that αp1 = ∞ if vp (x) = ∞. Let B = ⨁p∈ℙ B(p) and A the subgroup of B ⊕ G generated by all elements of the form p(αp1 , . . . , αpn , x)−(αp2 , . . . , αpn , x) or p(αpn , x)−x. Let H = (B⊕G)/A, and proceed as in the proof of Theorem 2.7.2. Then H/G󸀠 is a simply presented torsion 󸀠 󸀠 group with G󸀠 = (A + G)/A ≅ G and |x󸀠 |H p = vp (x) for every x ∈ G, its image x in G and p ∈ ℙ.

82 | 2 Simply presented groups Corollary 2.7.7 (Warfield [152, 154]). Let M be an Ulm matrix. Then there is a simply presented group Hx of torsion-free rank 1 containing an element of infinite order with Ulm matrix M. Proof. Let G = ⟨x⟩ for some x of infinite order, and write M = [mp,k ]. Define vp (apk x) = mp,k if a ∈ ℤ and p ∤ a. Construct Hx as in the theorem. The rest follows as before. We will find the groups Pα and Hx useful. Exercises. 1. (Walker [148]) Show that Pα /pα Pα ≅ ⨁β 0. 2. (Walker [148]) The Walker groups Pα are totally projective. 3. (Walker [149]) If α = β + δ, then pβ Pα ≅ Pδ .

2.8 Simply presented torsion groups By Lemma 2.6.1, the group ℤp (= the p-localization of ℤ at the prime p) is simply presented. Since a torsion group G is a direct sum of its p-torsion parts tp G ≅ G ⊗ ℤp , we may examine torsion groups locally, one prime at a time. The following result is needed; it shows that simple presentations are maintained under such localization: Lemma 2.8.1. Let G be a simply presented group and p a prime. Then Gp = G ⊗ ℤp is a simply presented group. Proof. We can write G = F/H, where F is a free group on a set X and H is a subgroup of F generated by elements of the form nx or mx − ny (where x, y ∈ X and m, n ∈ ℤ). Then Gp can be identified with Fp /Hp by Theorem 1.17.9. Letting Z be the free group on the underlying set X 󸀠 of Fp , we have a natural epimorphism Z → Fp , whose kernel is denoted by K. This yields an isomorphism f : Z/K → Fp , and we can write f (Hp ) = (K + L)/K for some subgroup L of Z which is generated by elements of the form nx or mx − ny (where x, y ∈ X 󸀠 and m, n ∈ ℤ). But then K + L is generated by elements of the same form since K is, following from the fact that Fp is simply presented (Corollary 2.6.2). Consequently, the group Gp ≅ (Z/K)/((K + L)/K) ≅ Z/(K + L) is simply presented. −1

By Lemmas 2.8.1 and 2.5.3, a torsion group is simply presented if and only if each of its p-torsion parts is simply presented. Thus, for the torsion case, it suffices to study simply presented p-groups; they were introduced by Crawley and Hales [14], who called them T-groups. Now let G be a simply presented p-group with (faithful) presentation G = ⟨X; Λ⟩. Then a partial order ≤ is defined in X by y≤x

if pn x = y for some nonnegative integer n.

2.8 Simply presented torsion groups | 83

This partial order satisfies the minimum condition since G is a p-group, and the set M of all minimal elements consists exactly of those x ∈ X satisfying the relation px = 0 in G. It is clear that in case G is a reduced p-group, ≤ also satisfies the maximum condition with those elements of height 0. For each z ∈ X, we define Xz = {x : x ∈ X and x ≥ z}. Then {Xz : z ∈ M} is a partition of X, so the next result is immediate. Lemma 2.8.2 (Crawley–Hales [15]). Let G = ⟨X; Λ⟩ be a simply presented p-group. Then G = ⨁z∈M ⟨Xz ⟩. Proposition 2.8.3 (Crawley–Hales [15]). Let G = ⟨X; Λ⟩ be a simply presented p-group, and let Y be a subset of X. Then the subgroup ⟨Y⟩ of G and the factor group G/⟨Y⟩ are simply presented. Specifically, ⟨Y⟩ ≅ ⟨Y 󸀠 , Λ1 ⟩, where Y 󸀠 = ⟨Y⟩ ∩ X, and Λ1 consists of those elements of Λ involving only elements of Y 󸀠 , and G/⟨Y⟩ ≅ ⟨X2 , Λ2 ⟩, where X2 = X \ ⟨Y⟩, and Λ2 is the set of relations in Λ involving only elements of X2 and px = 0 for all x ∈ X2 with px ∈ ⟨Y⟩. Proof. We write G = FX /H, where H is the subgroup of FX generated by all elements corresponding to the defining relations in Λ. Let X1 = ⟨Y⟩ ∩ X and H1 the subgroup of FX1 generated by all elements of the form px or px − y (x, y ∈ X1 ) which appear as generators in H. We will show that ⟨Y⟩ ≅ G1 = FX1 /H1 . Clearly, the correspondence x 󳨃→ x (x ∈ X1 ) gives rise to an epimorphism FX1 → ⟨Y⟩ ≅ (FY + H)/H whose kernel contains H1 , yielding an epimorphism ϕ : G1 → ⟨Y⟩. Now define Z = {z : z ∈ X \ X1 and pz = wz ∈ X1 } and Xz = {x ∈ X : x ≥ z} for each z ∈ Z. Letting Hz be any group isomorphic to ⟨Xz ⟩, we obtain an isomorphism ψz : ⟨Xz ⟩ → Hz . Then we let A = G1 ⊕ ⨁ H z z∈Z

and N the subgroup of A generated by all elements of the form (−wz , 0, . . . , 0, pψz (z), 0, . . . ) [which we write simply as pψz (z)−wz ] with z ∈ Z. To show that the sum G1 +N is direct, let a ∈ G1 ∩ N, say a = ∑ ri (pψzi (zi ) − wzi ) with ri ∈ ℤ and zi ∈ Z. Then a − ∑ ri wzi = ∑ ri pψzi (zi ) is an element of G1 ∩ ⨁z∈Z Hz = 0; thus, a = ∑ ri wzi and ∑ ri pψzi (zi ) = 0. Then for i = 1, . . . , n, we obtain ri pψzi (zi ) = 0; thus, ri wzi = ri pzi = 0, which yields a = 0. Now there is an epimorphism f : FX → A/N such that x 󳨃→ (x + H1 , 0, . . . ) + N if x ∈ X1 , x 󳨃→ (0, . . . , 0, ψz (x), 0, . . . ) + N if x ∈ Xz (z ∈ Z), and x 󳨃→ 0 whenever

84 | 2 Simply presented groups x ∈ X \ (X1 ∪ ⋃z∈Z Xz ). Since we may verify by taking cases that H ⊆ ker f , we obtain a map G = FX /H → A/N sending ⟨Y⟩ onto (G1 + N)/N ≅ G1 . This yields a map ψ : ⟨Y⟩ → G1 such that ϕψ = 1⟨Y⟩ ; thus, ⟨Y⟩ ≅ G1 . To prove the second assertion, let X2 = X \ ⟨Y⟩, and define H2 to be the subgroup of FX2 generated by all elements of the form px or px − y (x, y ∈ X2 ) which appear as generators in H, and by all elements px (x ∈ X2 ) whenever px ∈ ⟨Y⟩. Then x 󳨃→ x (x ∈ X2 ) gives rise to an epimorphism FX2 → G/⟨Y⟩ whose kernel contains H2 , which yields a map ϕ2 : G2 = FX2 /H2 → G/⟨Y⟩. Finally, x 󳨃→ x (x ∈ X2 ) and x 󳨃→ 0 (x ∈ X ∩ ⟨Y⟩) induce a map FX → G2 whose kernel contains H, which yields a map FX /H → G2 whose kernel contains ⟨Y⟩. Therefore, there is a map ψ2 : G/⟨Y⟩ → G2 such that ϕ2 ψ2 = 1G/⟨Y⟩ ; thus, G/⟨Y⟩ ≅ G2 . We now introduce a concept that generalizes the canonical form we saw in the Richman–Walker groups. Let p be a prime. Following Walker [149], we call a subset X of a group G a p-basis of G if 1. px ∈ X whenever x ∈ X such that px ≠ 0; 2. for every a ∈ G, there are unique integers rx with 0 ≤ rx < p (x ∈ X) such that a = ∑x∈X rx x. The notion of a p-basis was introduced by Crawley and Hales [14], who called it a T-basis. For example, X = {a, pa, p2 a, . . . , pn−1 a} is a p-basis of a cyclic group ⟨a⟩ of order pn , and X = {1, p, p2 , . . . } is a p-basis of ℤ. Notice that a p-basis X does not contain 0, and X = 0 is a p-basis of G = 0. Clearly, any basis of the ℤ/pℤ-vector space G/pG is a p-basis of G/pG. We have the following: Proposition 2.8.4 (Crawley–Hales [16]). Let G be a p-group and X a subset of G. Then G is simply presented with simple presentation G = ⟨X; Λ⟩ if and only if X is a p-basis of G. Proof. Suppose that G = ⟨X; Λ⟩ is simply presented. By construction of ⟨X; Λ⟩, condition (1) holds. To prove (2), let a ∈ G. Then we can write a = ∑ni=1 ri xi since X generates G. In fact, since pn x ∈ X for all x ∈ X and n ≥ 0 with pn x ≠ 0, we may assume 0 ≤ ri < p. To prove uniqueness, assume that a = ∑ni=1 ri xi = ∑ni=1 si xi with xi ∈ X and 0 ≤ ri , si < p. The claim being trivial for n = 1, let n > 1, and suppose the assertion is true for n − 1. We may assume that x1 is not p-divisible by any of the other elements xi . By Lemma 1.12.4, the zero map ⟨px1 , x2 , . . . , xn ⟩ → ℤ(p∞ ) extends to a homomorphism ϕ : ⟨x1 , x2 , . . . , xn ⟩ → ℤ(p∞ ) by sending x1 onto an element of order p,

2.8 Simply presented torsion groups | 85

obtaining ϕ(a) = r1 ϕ(x1 ) = s1 ϕ(x1 ). Then r1 = s1 , and we have ri = si (i ≠ 1) by the induction hypothesis, as desired. Therefore, X is a p-basis of G. Conversely, assume that G has a p-basis X = {xi : i ∈ I}. Let FX be the free group on X and H the subgroup of FX generated by all elements of the form pn x whenever pn x = 0 in G (x ∈ X) and pm x − y whenever pm x = y in G (x, y ∈ X). Then the epimorphism ϕ : FX → G induces the epimorphism f :

G󸀠 = FX /H ∑ λi xi + H

→ 󳨃→

G ∑ λi xi .

Let a ∈ ker f . The group G󸀠 is simply presented. Therefore, X is a p-basis of G󸀠 by the first part of this proof, and we can write a = ∑ λi xi + H with 0 ≤ λi < p. Since ∑ λi xi = 0 in G and X is a p-basis of G, we have λi = 0 by uniqueness of λi . Thus, ker f = 0, and we obtain G ≅ G󸀠 . Proposition 2.8.5. Let G = ⟨X; Λ⟩ be a simply presented p-group. Then we have the following: 1. If a = r1 x1 + ⋅ ⋅ ⋅ + rn xn , where x1 , . . . , xn are distinct elements of X and 0 < r1 , . . . , rn < p (see Proposition 2.8.4), then for every ordinal α, we have a ∈ pα G if and only if xi ∈ pα G for all i = 1, . . . , n. 2. If x ∈ X and |x|Gp > 0, then there exists a y ∈ X such that x = py. 3. The groups pα G and G/pα G are simply presented for all ordinals α. In particular, the divisible part D of G and G/D are simply presented. 4. If Y is a subset of X, then A = ⟨Y⟩ is a nice subgroup of G. 5. If G is a reduced group of length λ, then G is a direct sum of simply presented groups Gi (i ∈ I), such that if λ is a limit ordinal, each group Gi has length < λ, and if λ is a successor ordinal, each subgroup pλ−1 Gi is cyclic of order p. Proof. We prove that a ∈ pα G implies that xi ∈ pα G for all i = 1, . . . , n by transfinite induction on α. Suppose the assertion holds for all ordinals < α. It suffices to assume that α is a successor ordinal. Therefore, we write a = pb for some b = t1 y1 +⋅ ⋅ ⋅+tk yk ∈ pα−1 G for 0 < ti < p and distinct elements yi ∈ X for i = 1, . . . , k. By the induction hypothesis, we have y1 , . . . , yk ∈ pα−1 G. Then pb = t1 (py1 ) + ⋅ ⋅ ⋅ + tk (pyk ), and we rewrite this sum as pb = u1 z1 + ⋅ ⋅ ⋅ + ul zl , using distinct elements zi ∈ X ∩ pα G, and 0 < ui < p (i = 1, . . . , l). By uniqueness of the representation of a, we have {x1 , . . . , xn } = {z1 , . . . , zl } ⊆ pα G. To show (2), let x ∈ X and assume that x = pg for some g ∈ G. Since X is a p-basis of G (Proposition 2.8.4), we can write g = ∑i∈I ri xi with 0 < ri < p and xi ∈ X (i ∈ I). Then pxi ∈ X for all i ∈ J = {i ∈ I : pxi ≠ 0}, and the equation x = ∑i∈J ri pxi reduces to x = pxj for some j ∈ J. The third assertion follows from (1) and Proposition 2.8.3. To prove (4), let g ∈ G. By Proposition 2.8.4, we have g = r1 x1 + ⋅ ⋅ ⋅ + rm xm + s1 y1 + ⋅ ⋅ ⋅ + sn yn for elements xi ∈ X \ Y, yj ∈ Y, and 0 ≤ ri , sj < p (i = 1, . . . , m, j = 1, . . . , n). To

86 | 2 Simply presented groups show that b = r1 x1 + ⋅ ⋅ ⋅ + rm xm is an element of maximal p-height in the coset g + A, let a ∈ A, and write a = t1 y1󸀠 + ⋅ ⋅ ⋅ + tl yl󸀠 for some yi󸀠 ∈ Y and 0 < ti < p (i = 1, . . . , l). Then by (1), we have |b| = |r1 x1 | ∧ ⋅ ⋅ ⋅ ∧ |rm xm | ≥ |r1 x1 | ∧ ⋅ ⋅ ⋅ ∧ |rm xm | ∧ |t1 y1󸀠 | ∧ ⋅ ⋅ ⋅ ∧ |tl yl󸀠 | = |b + a|. Therefore, A is nice in G. To show (5), write G = ⨁z∈M ⟨Xz ⟩ as in Lemma 2.8.2, and let α be any ordinal. If pα ⟨Xz ⟩ has nonzero elements, then z ∈ pα ⟨Xz ⟩ by (1). Letting β = |z|p , we obtain pβ ⟨Xz ⟩ = ⟨z⟩; thus, the length of ⟨Xz ⟩ is β + 1, and the first assertion follows. If λ − 1 exists, let M 󸀠 = {z ∈ M : |z|p = λ−1}, pick z 󸀠 ∈ M 󸀠 , and define G1 = ⟨Xz 󸀠 ⟩⊕⨁z∈M\M 󸀠 ⟨Xz ⟩. Then G = G1 ⊕ ⨁z∈M 󸀠 \{z 󸀠 } ⟨Xz ⟩. Lemma 2.8.6 (Crawley–Hales [16]). Let G = ⟨X; Λ⟩ be a reduced simply presented p-group, and let X0 = {x ∈ X : |x|Gp = 0}. Then we have the following: 1. G = ⟨X0 ⟩. 2. The set {x + pG : x ∈ X0 } is a basis of G/pG. 3. If Y ⊆ X0 and a ∈ ⟨Y⟩ with |a|Gp > 0, then |a|⟨Y⟩ p > 0. Proof. To prove (1), let g ∈ G, and write g = ∑i∈I ri xi with ri ∈ ℤ, xi ∈ X (i ∈ I). We fix i ∈ I, and let y1 = xi . If |y1 |p ≠ 0, there is a y2 ∈ X such that y1 = py2 by Proposition 2.8.5(2), and in case |y2 |p ≠ 0, there is a y3 ∈ X with y2 = py3 , et cetera. Since any decreasing sequence of ordinals becomes stationary, we must have |yn |p = 0 for some n ∈ ℕ, i. e., yn ∈ X0 . Therefore, g ∈ ⟨X0 ⟩. To show (2), it suffices to verify that the subset {x +pG : x ∈ X0 } of G/pG is independent. Assume that ∑i∈I λi xi ∈ pG with 0 ≤ λi < p and xi ∈ X0 (i ∈ I). Then for each i ∈ I, we have λi = 0, since otherwise xi ∈ pG, by Proposition 2.8.5(1), is a contradiction. To prove the last assertion, let Y ⊆ X0 and a ∈ ⟨Y⟩ with |a|Gp > 0. Then a = ∑i∈I μi yi with 0 ≠ μi ∈ ℤ and yi ∈ Y. For each i ∈ I, we write μi = si p + ri with 0 ≤ ri < p. Then ∑i∈I ri yi ∈ pG. Therefore, if ri ≠ 0, then yi ∈ pG by Proposition 2.8.5(1), which is impossible. Thus, ri = 0 for all i ∈ I, and we obtain a ∈ p⟨Y⟩. The next result follows from Proposition 2.8.5(4), taking 𝒞 to be all ⟨Y⟩ where Y ⊆ X and G = ⟨X; Λ⟩: Proposition 2.8.7 (Crawley–Hales [14], Hill [47]). Simply presented p-groups have nice systems. Consequently, Hill’s Theorem 2.3.3 yields the following classification of simply presented torsion groups: Corollary 2.8.8 (Crawley–Hales [14]). Two simply presented torsion groups are isomorphic if and only if they have the same Ulm invariants. The following useful fact was extracted from [27, p. 99]:

2.8 Simply presented torsion groups | 87

Lemma 2.8.9. Let p be a prime, σ a limit ordinal, and B a direct sum of cyclic p-groups, say B ≅ ⨁i∈I ℤ/pni ℤ (ni ∈ ℕ). Let A = ⟨X; Λ⟩ be a reduced simply presented p-group of length σ, and let M be the set of all minimal elements in X. For every γ < σ, let Mγ = {x ∈ M : |x|p ≥ γ}, and assume that |Mγ | ≥ |I|. Then there exists a simply presented p-group C such that C/pσ C ≅ A and pσ C ≅ B. Proof. Let yi0 be a generator of ℤ/pni ℤ (i ∈ I), and write B = ⟨Y, ΛY ⟩, where Y = {yi0 , yi1 , . . . , yi(ni −1) : i ∈ I}, and ΛY consists of relations pyij = yi(j+1) with j = 0, . . . , ni − 2 and pyi(ni −1) = 0 (i ∈ I). As usual, we assume the presentation of A is faithful as well. For (1), we define a simply presented p-group C with presentation C = ⟨X 󸀠 ; Λ󸀠 ⟩, where X 󸀠 = X ∪ Y. We wish to define Λ󸀠 with the following properties: 1. If px = y in is Λ, then it is in Λ󸀠 ; 2. ΛY ⊆ Λ󸀠 ; 3. For every i ∈ I and γ < σ there is at least one x ∈ Mγ such that px = yi0 ∈ Λ󸀠 ; 4. For every x ∈ M there is a unique yi0 such that px = yi0 is in Λ󸀠 . First, for each γ < σ, we use induction on γ to construct injective functions ϕγ : Y0 = {yi0 : i ∈ I} → Mγ , with the property that if ϕα (yi0 ) ∈ Mγ for some α < γ, then ϕγ (yi0 ) = ϕα (yi0 ). Let γ < σ, and assume that suitable maps ϕα : Y0 → Mα are already constructed for all α < γ. We define ϕγ (yi0 ) for each yi0 ∈ Y0 . If ϕα (yi0 ) ∈ Mγ for some α < γ, let ϕγ (yi0 ) = ϕα (yi0 ). If not (that is, if |ϕα (yi0 )| < γ for all α < γ), let ϕγ (yi0 ) = x for some x ∈ Mγ in such a way that ϕγ is injective. This can be done since |Mγ | ≥ |I|. This completes the induction. Now construct Λ󸀠 to include the following: (a) all relations of the form px = y, where y ∈ Y0 and ϕγ (y) = x for some γ < σ; (b) one relation px = y for each x ∈ M that does not appear in any of the relations in (a), for some y ∈ Y0 ; (c) the relations defined by properties (1) and (2) above. We confirm that Λ󸀠 satisfies properties (1–4). By construction, we have (1) and (2) immediately; (3) because ϕγ is a function, and the existence of the yi0 in (4) by construction. It remains to prove yi0 is unique. If x ∈ ̸ ⋃α |x|Cp = α for some x ∈ X. Then in A, x = py for some y ∈ X with |y|Ap ≥ α by Proposition 2.8.5(2) applied to pα A. Then x = py in C as well, so |y|Cp < |x|Cp = α. From the induction hypothesis, we get |y|Cp = |y|Ap ≥ α, a contradiction. Now consider any yi0 . For every γ < σ there is an x ∈ X such that yi0 = px and |x|Cp = |x|Ap ≥ γ. Hence, |yi0 |Cp ≥ σ. Every element of Y is a multiple of some yi0 , so |y|Cp ≥ σ for all y ∈ Y. Since all of the relations are in the proper form, to prove this presentation is faithful, it suffices to prove that the elements of X 󸀠 are distinct and nonzero in C. If x ∈ X, then |x + K|p ≤ |x + H|p < σ shows that x ∉ K; thus, x ≠ 0 in C. Likewise, we see that the elements of X are distinct in C. Next suppose some y ∈ Y is 0 in C. Then y ∈ K, which is generated by expressions of the following forms: px − z where x, z ∈ X and px ≠ 0 in A; px = yi0 for some x ∈ X with px = 0 in A; any expression that reduces to 0 in B. Suppose there is a nonzero term of the form px − yi0 . Then there must be another term involving px to allow the sum to reduce to yi0 . But the only possibility is px − yj0 with j ≠ i, but this is impossible based on our construction of Λ󸀠 . Thus, the only terms involve relations in Y which say y = 0 in B. Likewise, the elements of Y are distinct in C since yi0 − yj0 ∈ K implies px = yi0 and px 󸀠 = yj0 are in Λ󸀠 for some x, x󸀠 ∈ X, which forces x = x󸀠 , which violates (4) unless yi0 = yj0 . Clearly, the elements of X and Y are distinct in C since their heights are different. Next we show that pσ C ≅ B. By Proposition 2.8.5(1), we have pσ C = ⟨X 󸀠󸀠 ⟩ with X 󸀠󸀠 = {x ∈ X 󸀠 : |x|Cp ≥ σ}. Then X 󸀠󸀠 = ⟨X 󸀠󸀠 ⟩ ∩ X 󸀠 , so Proposition 2.8.3 yields pσ C ≅ ⟨X 󸀠󸀠 , Λ1 ⟩, where Λ1 consists of those elements of Λ󸀠 involving only elements of X 󸀠󸀠 . Assume Λ󸀠 has a relation px = y (x, y ∈ X 󸀠󸀠 ); that is, x, y ∈ X 󸀠 = X ∪ Y with |x|Cp , |y|Cp ≥ σ. Then x ∈ X is impossible since X has no elements in C of p-height ≥ σ. Thus, px = y is one of the relations pyi0 = yi1 , . . . , pyi(ni −2) = yi(ni −1) for some i ∈ I. It follows that pσ C ≅ ⟨X 󸀠󸀠 , Λ1 ⟩ = ⟨Y, ΛY ⟩ = B. Likewise, Proposition 2.8.3 shows that C/pσ C ≅ ⟨X2 , Λ2 ⟩, where X2 = X 󸀠 \ pσ C = {x ∈ X 󸀠 : |x|Cp < σ}, and Λ2 consists of all relations in Λ󸀠 involving only elements of X2 and px = 0 for all x ∈ X2 with px ∈ pσ C. Then Λ2 = Λ, and we have C/pσ C ≅ ⟨X2 , Λ2 ⟩ ≅ ⟨X, Λ⟩ = A, as desired. Lemma 2.8.10. Let G be a p-group and σ a limit ordinal. If G/pσ G is simply presented such that pσ G is cyclic, then G is simply presented. σ

Proof. Write G/pσ G = ⟨X, Λ⟩, and pσ G = ⟨z⟩. Then |z|pp γ

G

= 0, so |z|Gp = σ by

Lemma 1.10.4(2). Let γ < σ. Since |z|pp G > 0, there is a y ∈ pγ G such that z = py. Since y ∈ ̸ pσ G, ȳ = y + pσ G is a nonzero element of G/pσ G, so ȳ = a1 x1 + ⋅ ⋅ ⋅ + an xn for distinct elements x1 , . . . , xn of X and 0 < ai < p for all 1 ≤ i ≤ n. Since ȳ ∈ pγ G/pσ G = pγ (G/pσ G) (Lemma 1.10.5), we have xi ∈ pγ (G/pσ G) for all i. Thus, ȳ ∈ ⟨Xγ ⟩, where σ

Xγ = {x ∈ X : |x|pG/p that

σ

|x|pG/p G

G

≥ γ}. Let Mγ be the set of all minimal elements x in X such

≥ γ. Then |Mγ | ≥ 1 since Xγ ≠ 0. Therefore, by Lemma 2.8.9, applied to

2.8 Simply presented torsion groups | 89

pσ G and G/pσ G, there is a simply presented p-group C, such that C/pσ C ≅ G/pσ G and pσ C ≅ pσ G. Since simply presented p-groups have nice systems (Proposition 2.8.7) and uα (C, pσ C) = uα (G, pσ G) for all ordinals α (Lemmas 1.13.6 and 1.13.10), we obtain C ≅ G by Theorem 2.3.2. As an application of Lemma 2.8.10, we have the following: Proposition 2.8.11. Generalized Prüfer groups are simply presented. Proof. The assertion is shown by transfinite induction on α. For n < ω, the claim is true since Hn ≅ ℤ/pn ℤ. Suppose α ≥ ω, and assume that the claim holds for all ordinals < α. If α is a limit ordinal, then Hα = ⨁βi xm󸀠 for all k ∈ I 󸀠 . If xn >i xm󸀠 , let m = m󸀠 . If xn >f xm󸀠 , then there is a positive integer r such that |pα rxn |p ≩ |pα rxm󸀠 |p is not true for any α < ω, which implies that xm󸀠 ≥ xn . Then for any k ∈ I 󸀠 , we have xk ≥ xm󸀠 ≥ xn or xk >i xm󸀠 ≥ xn . Thus, m = n yields xk ≥ xm or xk >i xm for all k ∈ I, as desired.

3.11 k-Subgroups and k-basic subgroups | 155

Lemma 3.11.4. Let C be a subgroup of G, and let x ∈ C be primitive in G. Assume that C = A ⊕ B and F = ⟨x1 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨xm󸀠 ⟩ ⊕ W are ∗-valuated coproducts in G such that F ⊆ C and W ⊆ G(M ∗ ), where ‖x1 ‖ ∼ ⋅ ⋅ ⋅ ∼ ‖xm󸀠 ‖, x1 , . . . , xm󸀠 are primitive in G, x1 = rx for some r ∈ ℤ with M = ‖a0 ‖ for some a0 ∈ A. Suppose that (a) δa0 = x1 + ⋅ ⋅ ⋅ + xm + w0 (δ ∈ ℤ, m ≤ m󸀠 , w0 ∈ W); (b) xi = ai + bi (ai ∈ A, bi ∈ B, i = 1, . . . , m) and kai ∈ F (i = 1, . . . , m) for some positive integer k. Then there is a primitive element g in G such that g ∈ A + W and ‖g‖ ∼ ‖x‖. In fact, if Λ is any finite (possibly empty) set of primes that do not divide r such that ‖x‖p ≤ ‖a0 ‖p for all p ∈ Λ, then after replacing x by a suitable nonzero multiple of itself, the primitive element g satisfies the following condition (∗) with respect to Λ and x: (∗) g = t(δa0 − w0 ) + g 󸀠 = c1 x1 + ⋅ ⋅ ⋅ + cm󸀠 xm󸀠 (t, c1 , . . . , cm󸀠 ∈ ℤ, c1 ≠ 0, g 󸀠 ∈ A + W), and for all p ∈ Λ, we have p ∤ t, ‖g 󸀠 ‖p ≥ ‖pg‖p , and ‖g‖p = ‖δa0 ‖p = ‖x‖p = ‖c1 x1 ‖p . If n is a positive integer and g is any element satisfying condition (∗) with respect to Λ and x, then ng satisfies condition (∗) with respect to Λ and nx. Proof. Since C = A ⊕ B is a ∗-valuated coproduct in G, and δa0 − (x1 + ⋅ ⋅ ⋅ + xm ) = w0 ∈ C ∩ G(M ∗ ), we have b1 + ⋅ ⋅ ⋅ + bm ∈ G(M ∗ ). By (b), we can write m󸀠

kai = ∑ cij xj + wi j=1

(cij ∈ ℤ, wi ∈ W)

for i = 1, . . . , m. Then k(b1 + ⋅ ⋅ ⋅ + bm ) ∈ G(M ∗ ) ∩ F = W by Corollary 3.10.4, and we m󸀠 m m󸀠 obtain k(x1 + ⋅ ⋅ ⋅ + xm ) = ∑m i=1 (∑j=1 cij xj + wi + kbi ) = ∑i=1 ∑j=1 cij xj + w for some w ∈ W. Then a comparison of coefficients of xj yields k = c1j + ⋅ ⋅ ⋅ + cmj for j = 1, . . . , m. Now let Λ be a finite set of primes not dividing r such that ‖x‖p ≤ ‖a0 ‖p for all p ∈ Λ. Clearly, g = δa0 − w0 satisfies condition (∗), but g is not necessarily primitive. Let P be the set of all primes p such that ‖pα x1 ‖p = ⋅ ⋅ ⋅ = ‖pα xm󸀠 ‖p for some α < ω, and notice that ℙ \ P is a finite set. Then for any element g = t(δa0 − w0 ) + g 󸀠 = c1 x1 + ⋅ ⋅ ⋅ + cm󸀠 xm󸀠

(t, c1 , . . . , cm󸀠 ∈ ℤ, c1 ≠ 0, g 󸀠 ∈ A + W)

satisfying condition (∗), let Q(g) = {p ∈ (ℙ \ P) ∪ Λ : ‖g‖p = ‖ci xi ‖p for some i ≤ m󸀠 }. Notice that Q(g) contains the set Λ, and that for any positive integer n, ng = t(δna0 − nw0 ) + ng 󸀠 = c1 nx1 + ⋅ ⋅ ⋅ + cm󸀠 nxm󸀠

156 | 3 Warfield groups satisfies condition (∗) with respect to Λ and nx since ‖ng 󸀠 ‖p ≥ ‖png‖p and ‖ng‖p = ‖δna0 ‖p = ‖nx‖p = ‖c1 nx1 ‖p for all p ∈ Λ. Thus, the last statement in the lemma follows. Therefore, we have Q(g) ⊆ Q(ng), and we can replace x by a nonzero multiple if necessary. If ℙ \ P ⊆ Q(g), then g is primitive (Lemma 3.10.11) as desired. Therefore, assume that ℙ \ P ⊈ Q(g). Letting q ∈ ℙ \ P with q ∉ Q(g), we will show that there exists a g in G, satisfying condition (∗), such that {q} ∪ Q(g) ⊆ Q(g). Then finitely many applications of this result will produce an element g ∗ ∈ G satisfying condition (∗), such that ℙ \ P ⊆ Q(g ∗ ). Hence, g ∗ is primitive. Since q ∉ Q(g), we have ‖g‖q ≠ ‖ci xi ‖q for i = 1, . . . , m󸀠 . Therefore, by Lemma 3.11.3, there exists a nonempty subset J of I = {1, . . . , m} and a γ < ω such that ‖qγ xl ‖q = ‖qγ xr ‖q for all l, r ∈ J and |qα xi |q ≩ |qα xl |q for infinitely many α < ω whenever l ∈ J and i ∈ I \ J. By replacing x with qγ x, we may assume that ‖xl ‖q = ‖xr ‖q for all l, r ∈ J. Let k = qj k 󸀠 with q ∤ k 󸀠 . Then we have qj+1 |cil

whenever l ∈ J and i ∈ I \ J

since qj+1 ∤ cil yields |qα+j xl |q ≥ |qα cil xl |q ≥ |qα kai |q = |qα+j ai |q ≥ |qα+j xi |q for all α < ω, which is impossible. For all l, r ∈ J, we have ‖kal ‖q ≤ ‖clr xr ‖q = ‖clr xl ‖q ≤ ‖clr al ‖q ; j+1 hence, qj |clr . Now fix an r ∈ J. Since k = ∑m i=1 cir and q |cir for all i ∈ I \ J, there exists an l ∈ J such that qj+1 ∤ clr . Then ‖qj xl ‖q = ‖kxl ‖q ≤ ‖kal ‖q ≤ ‖kal − wl ‖q ≤ ‖clr xr ‖q = ‖qj xr ‖q = ‖qj xl ‖q . Thus,

‖kxl ‖q = ‖kal ‖q = ‖kal − wl ‖q = ‖clr xr ‖q . Since clr ≠ 0, we have ‖kal − wl ‖ ∼ ‖x‖ ∼ ‖g‖. Therefore, there are positive integers c and β with q ∤ c such that ‖qβ g‖q ≥ ‖q(kal − wl )‖q

and ‖c(kal − wl )‖p ≥ ‖pg‖p for all p ∈ Q(g).

Let g = qβ g + c(kal − wl ) = c1 x1 + ⋅ ⋅ ⋅ + cm󸀠 xm󸀠 with ci = qβ ci + ccli (i = 1, . . . , m󸀠 ). It is clear that β can be replaced by a larger integer, so since c1 ≠ 0, we can assume that c1 ≠ 0 and have ‖g‖ ∼ ‖x‖. Since ‖qβ g‖q > ‖c(kal − wl )‖q , ‖qβ cr xr ‖q ≥ ‖qβ g‖q , and ‖cclr xr ‖q = ‖kal − wl ‖q < ‖q(kal − wl )‖q ≤ ‖qβ g‖q , we obtain ‖g‖q = ‖kal − wl ‖q = ‖clr xr ‖q = ‖qβ cr xr ‖q ∧ ‖cclr xr ‖q = ‖cr xr ‖q . To verify that g satisfies condition (∗), we write g = qβ t(δa0 − w0 ) + g 󸀠 = c1 x1 + ⋅ ⋅ ⋅ + cm󸀠 xm󸀠

3.11 k-Subgroups and k-basic subgroups | 157

with g 󸀠 = qβ g 󸀠 + c(kal − wl ) ∈ A + W, and assume that p ∈ Λ. Then p ∤ qβ t, and 󵄩󵄩 󸀠 󵄩󵄩 󵄩 󵄩 β 󵄩 󵄩 󵄩 󵄩 β 󸀠󵄩 󵄩 󵄩󵄩g 󵄩󵄩p ≥ 󵄩󵄩󵄩q g 󵄩󵄩󵄩p ∧ 󵄩󵄩󵄩c(kal − wl )󵄩󵄩󵄩p ≥ ‖pg‖p = 󵄩󵄩󵄩pq g 󵄩󵄩󵄩p ∧ 󵄩󵄩󵄩pc(kal − wl )󵄩󵄩󵄩p = ‖pg‖p . By assumption, we have ‖g‖p = ‖δa0 ‖p = ‖x‖p = ‖c1 x1 ‖p ; hence, 󵄩 󵄩 󵄩 󵄩 ‖g‖p = 󵄩󵄩󵄩qβ g 󵄩󵄩󵄩p ∧ 󵄩󵄩󵄩c(kal − wl )󵄩󵄩󵄩p = ‖g‖p m and ‖ccl1 x1 ‖p ≥ ‖ ∑i=1 ccli xi ‖p = ‖c(kal − wl )‖p ≥ ‖pg‖p yield 󸀠

󵄩 󵄩 ‖g‖p = ‖δa0 ‖p = ‖x‖p = ‖c1 x1 ‖p = 󵄩󵄩󵄩qβ c1 x1 󵄩󵄩󵄩p ∧ ‖ccl1 x1 ‖p = ‖c1 x1 ‖p . Consequently, g satisfies condition (∗). It remains to show that {q} ∪ Q(g) ⊆ Q(g). Clearly, q ∈ Q(g) since ‖g‖q = ‖cr xr ‖q . If p ∈ Q(g), then ‖g‖p = ‖ci xi ‖p for some i ≤ m󸀠 . Therefore, since ‖ccli xi ‖p ≥ ‖c(kal − wl )‖p ≥ ‖pg‖p , we have ‖g‖p = ‖g‖p = ‖ci xi ‖p = ‖ci xi ‖p ∧ ‖ccli xi ‖p = ‖ci xi ‖p . Therefore, p ∈ Q(g) as required. Corollary 3.11.5. Let C be a k-basic subgroup of G such that C = A ⊕ B is a ∗-valuated coproduct in G, and let a0 ∈ A with M = ‖a0 ‖. Assume that δa0 = x1 + ⋅ ⋅ ⋅ + xm + w0 for some positive integer δ, where ⟨x1 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨xm ⟩ ⊕ ⟨w0 ⟩ ⊆ C is a ∗-valuated coproduct in G with ‖x1 ‖ ∼ ⋅ ⋅ ⋅ ∼ ‖xm ‖ ∼ M, x1 , . . . , xm primitive in G, and w0 ∈ G(M ∗ ). Then there is a primitive element g in G such that ‖g‖ ∼ M and g ∈ (A + G(M ∗ )) ∩ C. If g is any such element in G, then g = a + b󸀠 (a ∈ A, b󸀠 ∈ B ∩ G(M ∗ )) and a is primitive in G with ‖na‖ = ‖ng‖ for some positive integer n. Proof. For i ∈ {1, . . . , m}, we write xi = ai + bi (ai ∈ A, bi ∈ B), and have ‖ai ‖ ≥ ‖xi ‖ ≥ ‖δa0 ‖ ≥ M. By Proposition 3.11.2(2), there is a positive integer k such that k⟨a1 , . . . , am ⟩ ⊆ ⟨Z⟩, where Z = X ∪̇ Y ⊆ C is a finite ∗-decomposition set in G satisfying {x1 , . . . , xm } ⊆ X, ‖x‖ ∼ M for all x ∈ X and Y ⊆ G(M ∗ ). Using Proposition 3.11.2(2) again, we obtain a finite ∗-decomposition set Z 󸀠 in G with Z ⊆ Z 󸀠 ⊆ C and containing a set Y 󸀠 = {y1 , . . . , ys } with lw0 = l1 y1 + ⋅ ⋅ ⋅ + ls ys for some nonzero integers l, l1 , . . . , ls . Let i ∈ {1, . . . , s}. Since lw0 ∈ G(M ∗ ), we have li yi ∈ G(M ∗ ). Thus, ‖yi ‖ ≁ M since yi is primitive (Proposition 3.10.3(1)). Then W = ⟨Y ∪ Y 󸀠 ⟩ is contained in G(M ∗ ), and we obtain the ∗-valuated coproduct F = ⟨X⟩ ⊕ W with F ⊆ C, lw0 ∈ W, δla0 = lx1 + ⋅ ⋅ ⋅ + lxm + lw0 ∈ F, and k⟨la1 , . . . , lam ⟩ ⊆ F. By Lemma 3.11.4 applied to lx1 , . . . , lxm , there is a primitive element g ∈ A + W ⊆ (A + G(M ∗ )) ∩ C with ‖g‖ ∼ ‖lx1 ‖ ∼ M. This proves such a g exists.

158 | 3 Warfield groups Now let g be such an element of G, and write g = a+b󸀠 with a ∈ A and b󸀠 ∈ B. Then b is an element of A + G(M ∗ ) since g is, so we have b󸀠 = a󸀠 + g 󸀠 for some a󸀠 ∈ A and g 󸀠 ∈ G(M ∗ ); hence, b󸀠 ∈ G(M ∗ ). Since C is k-basic, there is a finite ∗-decomposition set X 󸀠 in G such that X 󸀠 ⊆ C and rb󸀠 ∈ ⟨X 󸀠 ⟩ for some positive integer r. If rb󸀠 = 0, then rg = ra. Hence, a is primitive in G, and the assertion follows. Assuming that b󸀠 has infinite order, we obtain rb󸀠 = r1 z1 +⋅ ⋅ ⋅+rt zt for some z1 , . . . , zt ∈ X 󸀠 and nonzero integers r1 , . . . , rt . Letting z ∈ {r1 z1 , . . . , rt zt }, we have z ∈ G(M ∗ ). Hence, ‖z‖ ≁ M since z is primitive. Then ‖z‖ ≥ ‖rb󸀠 ‖ ≥ ‖rg‖ and ‖z‖ ≁ ‖g‖ yield z ∈ G(‖rg‖∗ ); thus, rb󸀠 ∈ G(‖rg‖∗ ). By Proposition 3.10.5, ra ∈ rg + G(‖rg‖∗ ) is primitive in G and ‖na‖ = ‖ng‖ for some positive integer n. 󸀠

Corollary 3.11.6 (Hill–Megibben [53]). Let A ⊕ B be a ∗-valuated coproduct in a group G such that C = A ⊕ B is a k-basic subgroup of G, and A has finite torsion-free rank. Then there is a finite ∗-decomposition set X in G such that X ⊆ A and A/⟨X⟩ is torsion. Proof. We show the assertion by induction on n = r0 (A). For n = 0, simply let X = 0. Now suppose that n > 0, and that the claim holds for all k < n. Since C is a k-subgroup of G, Lemma 3.5.1 shows that every increasing sequence of compatibility classes of Ulm matrices realized by nonzero elements of A of infinite order becomes stationary. Thus, there is an Ulm matrix M and a nonzero element a0 ∈ A of infinite order such that ‖a0 ‖ = M and A ∩ G(M ∗ ) is torsion; hence, a0 ∉ G(M ∗ ). Since C is k-basic in G, we have δa0 = x1 + ⋅ ⋅ ⋅ + xm + w0 for some positive integer δ, where ⟨x1 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨xm ⟩ ⊕ ⟨w0 ⟩ ⊆ C is a ∗-valuated coproduct in G, each element xi is primitive in G with ‖xi ‖ ∼ M and w0 ∈ G(M ∗ ). By Corollary 3.11.5, there is a primitive element g in G such that g ∈ C and if g = a + b󸀠 (a ∈ A, b󸀠 ∈ B), then a is primitive in G. Since C is k-basic in G, there is a ∗-valuated coproduct C1 = ⟨a⟩ ⊕ D1 in G with C/C1 finite. Let A󸀠 = A∩D1 . Then C 󸀠 = ⟨a⟩⊕A󸀠 ⊕B is a ∗-valuated coproduct in G with C/C 󸀠 finite, and by Proposition 3.11.2(1), C 󸀠 is k-basic in G. We have r0 (A󸀠 ) < r0 (A). Therefore, by the induction hypothesis, there is a finite ∗-decomposition set X 󸀠 in G such that X 󸀠 ⊆ A󸀠 with A/⟨X 󸀠 ⟩ torsion. Then X = {a} ∪ X 󸀠 is a ∗-decomposition set in G, and A/⟨X⟩ is torsion, as desired. Theorem 3.11.7 (Hill–Megibben [53]). Let A ⊕ B be a ∗-valuated coproduct in a group G such that C = A ⊕ B is a k-basic subgroup of G. Suppose that x ∈ C is a primitive element in G. Then there is a ∗-valuated coproduct C 󸀠 = ⟨x󸀠 ⟩ ⊕ ⟨y⟩ ⊕ A󸀠 ⊕ B󸀠 in G with A󸀠 ⊆ A, B󸀠 ⊆ B and C 󸀠 ⊆ C such that 1. r0 (A/A󸀠 ) ≤ 1, r0 (B/B󸀠 ) ≤ 1, and C/C 󸀠 is finite; 2. x󸀠 is a nonzero multiple of x, and either y = 0 or y is primitive in G with ‖x‖ ∼ ‖y‖.

3.11 k-Subgroups and k-basic subgroups | 159

Proof. We write x = a0 + b0 with a0 ∈ A and b0 ∈ B, and let M = ‖a0 ‖. By Proposition 3.11.2(2), there is a positive integer δ and an r ∈ ℤ such that δa0 = rx + x2 + ⋅ ⋅ ⋅ + xm + w0 , where ⟨x⟩ ⊕ ⟨x2 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨xm ⟩ ⊕ ⟨w0 ⟩ ⊆ C is a ∗-valuated coproduct in G with each xi primitive, ‖xi ‖ ∼ ‖x‖, and w0 ∈ G(M ∗ ). Letting s = δ − r, we have δb0 = sx − (x2 + ⋅ ⋅ ⋅ + xm + w0 ). Case I: s = 0. Then δ = r; therefore, we have δa0 = δx+x2 +⋅ ⋅ ⋅+xm +w0 . Now ‖δa0 ‖ ≤ ‖δx‖ and ‖x‖ ≤ ‖a0 ‖ yield ‖δa0 ‖ = ‖δx‖. Hence, Proposition 3.10.6 shows that δa0 is primitive in G. Since C is k-basic in G, there is a ∗-valuated coproduct C1 = ⟨δa0 ⟩ ⊕ C2 in G with C1 ⊆ C and C/C1 finite. Let A󸀠 = A ∩ C2 . Then A ∩ C1 = ⟨δa0 ⟩ ⊕ A󸀠 ⊆ ⟨δa0 ⟩ ⊕ C2 is a ∗-valuated coproduct in G; thus, C 󸀠 = ⟨δa0 ⟩ ⊕ A󸀠 ⊕ B is a ∗-valuated coproduct in G. Since A/(A ∩ C1 ) ≅ (A + C1 )/C1 ⊆ C/C1 is finite, C/C 󸀠 = (A ⊕ B)/(⟨δa0 ⟩ ⊕ A󸀠 ⊕ B) is finite, and we have r0 (A/A󸀠 ) = r0 (C/(A󸀠 ⊕ B)) = r0 (C 󸀠 /(A󸀠 ⊕ B)) + r0 (C/C 󸀠 ) = 1. We use Proposition 3.10.6 again and obtain the ∗-valuated coproduct C 󸀠 = ⟨δx⟩⊕A󸀠 ⊕B. Letting y = 0 and B󸀠 = B, the assertion follows. Case II: r = 0. Then δ = s; hence, δb0 = δx − (x2 + ⋅ ⋅ ⋅ + xm + w0 ), and we have ‖δb0 ‖ = ‖δx‖. Using a similar argument as above, we obtain a subgroup B󸀠 of B with r0 (B/B󸀠 ) = 1 such that C 󸀠 = ⟨δx⟩ ⊕ A ⊕ B󸀠 is a ∗-valuated coproduct in G with C/C 󸀠 finite, and we let y = 0 and A󸀠 = A. Case III: r ≠ 0 and s ≠ 0. Then ‖x‖ ≤ M ≤ ‖rx‖ shows that ‖x‖ ∼ M. If M ∼ ∞, we can proceed as in Case I since a0 is primitive by Section 3.10(b) and ‖nx‖ = ‖na0 ‖ = ∞ for some positive integer n. Now suppose that M ≁ ∞. By replacing x with a nonzero multiple, we may assume that gcd(δ, r) = 1, and therefore, gcd(r, s) = 1. Let x1 = rx, and write xi = ai + bi (ai ∈ A, bi ∈ B) for i = 1, . . . , m. Then ‖ai ‖ ≥ ‖xi ‖ ≥ M for all i = 1, . . . , m. By Proposition 3.11.2(2), there is a positive integer k such that k⟨a1 , . . . , am ⟩ ⊆ ⟨Z⟩, where Z = X ∪̇ Y ⊆ C is a finite ∗-decomposition set in G with X = {x1 , . . . , xm󸀠 } for some m󸀠 ≥ m, ‖xi ‖ ∼ M for all i = 1, . . . , m󸀠 , and Y ⊆ G(M ∗ ). Using Proposition 3.11.2(2) again, we can find a ∗-valuated coproduct F = ⟨x1 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨xm󸀠 ⟩ ⊕ W in G such that F ⊆ C, Y ⊆ W ⊆ G(M ∗ ), and lw0 ∈ W for some positive integer l (see the first part of the proof of Corollary 3.11.5). Then δla0 = lx1 + ⋅ ⋅ ⋅ + lxm + lw0 ∈ F, and by replacing x with lx, we may assume that l = 1. Now let Λ be any finite set of primes not dividing r. Then by Lemma 3.11.4, after replacing x by a suitable nonzero multiple, there is a primitive element g = t(δa0 − w0 ) + g 󸀠

160 | 3 Warfield groups with g 󸀠 ∈ A + W and ‖g‖ ∼ ‖x‖ such that p ∤ t, ‖g‖p = ‖x‖p , and ‖g 󸀠 ‖p ≥ ‖pg‖p for all p ∈ Λ. Since C is k-basic in G, we have a ∗-valuated coproduct C1 = ⟨g⟩ ⊕ D1 in G with C1 ⊆ C and C/C1 finite, so there are integers d ≠ 0 and λ such that dδa0 = λg + u (u ∈ D1 ), and dw0 ∈ C1 . Replacing x by a nonzero multiple, we can assume that gcd(d, λ) = 1 (see the last statement in Lemma 3.11.4), and have tλg+dg 󸀠 = t(dδa0 −u)+dg−dt(δa0 −w0 ) = dg + t(dw0 − u). If dw0 = d󸀠 g + y for some nonzero integer d󸀠 and y ∈ D1 , then d󸀠 g ∈ G(M ∗ ), which is impossible since d󸀠 g is primitive (Proposition 3.10.3(1)). Thus, dw0 ∈ D1 . Then for any p ∈ Λ with p|λ, we obtain ‖x‖p = ‖g‖p = ‖dg‖p ≥ ‖dg‖p ∧ ‖t(dw0 − u)‖p . Thus, 󵄩 󵄩 󵄩 󵄩 ‖x‖p ≥ 󵄩󵄩󵄩tλg + dg 󸀠 󵄩󵄩󵄩p ≥ ‖tλg‖p ∧ 󵄩󵄩󵄩dg 󸀠 󵄩󵄩󵄩p ≥ ‖pg‖p = ‖px‖p , which shows that |x|p = ∞. Consequently, we have p ∤ λ and ‖x‖p = ‖g‖p

whenever p ∈ Λ and |x|p ≠ ∞.

Likewise, for any finite set Λ󸀠 of primes not dividing s, there exists a primitive element h = t 󸀠 (δb0 − w0 ) + h󸀠 with h󸀠 ∈ B + W 󸀠 for some W 󸀠 ⊆ G(‖b0 ‖∗ ) and ‖h‖ ∼ ‖x‖ such that p ∤ t 󸀠 , ‖h‖p = ‖x‖p , and ‖h󸀠 ‖p ≥ ‖ph‖p for all p ∈ Λ󸀠 . Then there is a ∗-valuated coproduct C2 = ⟨h⟩ ⊕ D2 in G with C2 ⊆ C and C/C2 finite. Therefore, there are integers e ≠ 0 and μ such that eδb0 = μh + v

(v ∈ D2 ),

and after replacing x with a nonzero multiple if necessary, we have p ∤ μ and ‖x‖p = ‖h‖p

whenever p ∈ Λ󸀠 and |x|p ≠ ∞.

Now let Λ = {p ∈ ℙ : p | s}, and Λ󸀠 = {p ∈ ℙ : p | λ or ‖g‖p ≠ ‖x‖p } ∩ {p ∈ ℙ : |x|p ≠ ∞}. Notice that the primes in Λ do not divide r. Clearly, Λ and Λ󸀠 are disjoint; thus, the primes in Λ󸀠 do not divide s. If λ ≠ 0, then Λ󸀠 is finite since ‖g‖ ∼ ‖x‖. If λ = 0, we write ℙ = {pi : i < ω} and, if possible, replace Λ by Λn = Λ ∪ {pi : pi ∤ r and i ≤ n} for sufficiently large n < ω such that the resulting primitive element gn satisfies dn δa0 = λn gn + un

(dn ∈ ℤ \ {0}, un ∈ D1 )

with λn ≠ 0. In this case, we replace λ by λn and g by gn to obtain a finite Λ󸀠 . If λn = 0 for all n < ω, we have |x|p = ∞ for all p ∈ Λn (n < ω), that is, for all primes p with p ∤ r. But then Λ need not be replaced since Λ󸀠 is finite.

3.11 k-Subgroups and k-basic subgroups | 161

Next we will show that ‖eλg‖p ‖deδx‖p = { ‖dμh‖p

if p ∉ Λ󸀠 ,

if p ∈ Λ󸀠 .

Since ‖deδx‖ = ‖deδa0 ‖ ∧ ‖deδb0 ‖ ≤ ‖eλg‖ ∧ ‖dμh‖, the claim follows for primes p with |x|p = ∞. Suppose that |x|p ≠ ∞. If p ∈ Λ󸀠 , then p ∤ μ and ‖x‖p = ‖h‖p ; thus, ‖deδx‖p ≤ ‖dμh‖p = ‖dx‖p ≤ ‖deδx‖p . If p ∉ Λ󸀠 , then p ∤ λ and ‖g‖p = ‖x‖p , which implies that ‖deδx‖p ≤ ‖eλg‖p = ‖eg‖p ≤ ‖deδx‖p . Therefore, the claim holds in both cases. Now we write g = a + b󸀠 for a ∈ A and b󸀠 ∈ B. By Corollary 3.11.5, a is primitive in G, b󸀠 ∈ G(M ∗ ), and ‖na‖ = ‖ng‖ for some positive integer n. By replacing x with a nonzero multiple and using the last statement in Lemma 3.11.4, we may assume that ‖a‖ = ‖g‖. Similarly, if h = a󸀠 + b (a󸀠 ∈ A, b ∈ B), then b is primitive in G, and we can assume that ‖b‖ = ‖h‖. By replacing x with a nonzero multiple, we can assume that b󸀠 ∈ C1 . Since C1 = ⟨g⟩ ⊕ D1 is a ∗-valuated coproduct, and g is primitive with ‖g‖ ∼ M, we have b󸀠 ∈ D1 . Likewise, we can assume that a󸀠 ∈ D2 . Since a = g − b󸀠 with ‖a‖ = ‖g‖ and b = h − a󸀠 with ‖b‖ = ‖h‖, we obtain ∗-valuated coproducts C1 = ⟨a⟩ ⊕ D1 and C2 = ⟨b⟩ ⊕ D2 by Proposition 3.10.6. Now let A󸀠 = A ∩ D1 and B󸀠 = B ∩ D2 . Then C0 = ⟨a⟩ ⊕ ⟨b⟩ ⊕ A󸀠 ⊕ B󸀠 is a ∗-valuated coproduct in G, and since the groups C/C1 = (A ⊕ B)/(⟨a⟩ ⊕ D1 ) and C/C2 = (A ⊕ B)/(⟨b⟩ ⊕ D2 ) are finite, C/C0 ≅ A/(⟨a⟩ ⊕ A󸀠 ) ⊕ B/(⟨b⟩ ⊕ B󸀠 ) is finite. Further, we have r0 (A/A󸀠 ) = r0 ((⟨a⟩ ⊕ A󸀠 )/A󸀠 ) + r0 (A/(⟨a⟩ ⊕ A󸀠 )) = 1, and similarly, r0 (B/B󸀠 ) = 1. Since dδa0 = λg + u = λ(a + b󸀠 ) + u, we can write dδa0 = λa + u󸀠 with u󸀠 = λb󸀠 + u ∈ D1 ∩ A = A󸀠 . Likewise, we have eδb0 = μb + v󸀠 with v󸀠 = μa󸀠 + v ∈ B󸀠 . Then deδx = e(dδa0 ) + d(eδb0 ) = eλa + dμb + (eu󸀠 + dv󸀠 ). Thus, for any prime p, we have ‖eλa + dμb‖p ≥ ‖deδx‖p . If p ∉ Λ󸀠 , then ‖deδx‖p = ‖eλg‖p = ‖eλa‖p , and if p ∈ Λ󸀠 , we have ‖deδx‖p = ‖dμh‖p = ‖dμb‖p . In any case, we obtain ‖eλa + dμb‖p ≥ ‖deδx‖p ≥ ‖eλa + dμb‖p . Thus, ‖deδx‖ = ‖eλa + dμb‖. Letting z = eλa + dμb, we have ‖z‖ = ‖deδx‖; thus, ‖z‖p = ‖eλa‖p whenever p ∉ Λ󸀠 , and ‖z‖p = ‖dμb‖p if p ∈ Λ󸀠 . Notice that ‖x‖ ∼ M ≁ ∞ implies that |x|p ≠ ∞ for some prime p. In the case where λ = 0, we have p ∈ Λ󸀠 ; thus p ∤ μ, which shows that μ ≠ 0. Consequently, z ≠ 0 in any case. Then Lemma 3.10.11 shows that z is primitive in G since ‖a‖ ∼ ‖b‖. By Theorem 3.10.14, there is a primitive element y in G such that y ∈ ⟨a⟩ ⊕ ⟨b⟩ and ⟨z⟩ ⊕ ⟨y⟩ is a ∗-valuated coproduct in G with (⟨a⟩ ⊕ ⟨b⟩)/(⟨z⟩ ⊕ ⟨y⟩) finite. Then ‖y‖ ∼ ‖a‖ ∼ ‖x‖ and C 󸀠 = ⟨z⟩ ⊕ ⟨y⟩ ⊕ A󸀠 ⊕ B󸀠 is a ∗-valuated coproduct in G with C/C 󸀠 finite. Letting x󸀠 = deδx, we have ‖x󸀠 ‖ = ‖z‖ and x󸀠 = z + (eu󸀠 + dv󸀠 ); hence, C 󸀠 = ⟨x󸀠 ⟩ ⊕ ⟨y⟩ ⊕ A󸀠 ⊕ B󸀠 is a ∗-valuated coproduct in G by Proposition 3.10.6. This completes the proof.

162 | 3 Warfield groups The next result follows from the proof of [53, Corollary 2.5]: Corollary 3.11.8. Let A ⊕ B be a ∗-valuated coproduct in a group G, and X = {x1 , x2 , . . . } a countable ∗-decomposition set in G. Let C = ⟨X⟩, A0 = A ∩ C, B0 = B ∩ C, and C0 = A0 ⊕B0 . If C/C0 is torsion, then for every positive integer n, there exists a nonzero multiple xn󸀠 of xn and subgroups An , Bn , and Cn of C such that 1. Cn = ⟨x1󸀠 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨xn󸀠 ⟩ ⊕ An ⊕ Bn is a ∗-valuated coproduct in G; 2. Cn ⊆ Cn−1 and C0 /Cn is finite; 3. An ⊆ An−1 and A0 /An has torsion-free rank ≤ n. Proof. By Corollary 3.11.1 and Proposition 3.11.2(1), C0 = A0 ⊕ B0 is k-basic in G. We will show the assertion by induction, so first assume that n = 1. Then Theorem 3.11.7 yields a ∗-valuated coproduct C1 = ⟨x1󸀠 ⟩ ⊕ A1 ⊕ B1 in G (with B1 = ⟨y⟩ ⊕ B󸀠 as in Theorem 3.11.7) such that x1󸀠 is a nonzero multiple of x1 , C1 ⊆ C0 with C0 /C1 finite, and A1 ⊆ A0 with r0 (A0 /A1 ) ≤ 1. Now let n be an integer > 1. Assuming that the claim holds for all k < n, there 󸀠 is a ∗-valuated coproduct Cn−1 = ⟨x1󸀠 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨xn−1 ⟩ ⊕ An−1 ⊕ Bn−1 in G with nonzero 󸀠 multiples xi of xi (i = 1, . . . , n − 1) such that C0 /Cn−1 is finite and r0 (A0 /An−1 ) ≤ n − 1. By Proposition 3.11.2(3), D = An−1 ⊕ Bn−1 is k-basic in G. Since Cn−1 contains a nonzero 󸀠 multiple x̃n of xn , we can write x̃n = y + z, where y ∈ ⟨x1󸀠 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨xn−1 ⟩ and z ∈ D. Hence, ‖x̃n ‖ ≤ ‖y‖. Then ‖z‖ = ‖x̃n ‖ ∧ ‖y‖ = ‖x̃n ‖. Thus, z is primitive in G by Proposition 3.10.6. Using Theorem 3.11.7, we obtain a positive integer m and a ∗-valuated coproduct D󸀠 = ⟨mz⟩ ⊕ An ⊕ Bn in G such that An ⊆ An−1 with r0 (An−1 /An ) ≤ 1, and D󸀠 ⊆ D with D/D󸀠 󸀠 finite. Now let xn󸀠 = mx̃n . Then xn󸀠 = my + mz ∈ ⟨x1󸀠 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨xn−1 ⟩ ⊕ An ⊕ Bn ⊕ ⟨mz⟩, and 󸀠 we have ‖xn ‖ = ‖mz‖. Thus Proposition 3.10.6 shows that 󸀠 ⟩ ⊕ ⟨xn󸀠 ⟩ ⊕ An ⊕ Bn Cn = ⟨x1󸀠 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨xn−1 󸀠 is a ∗-valuated coproduct in G. Notice that Cn = ⟨x1󸀠 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨xn−1 ⟩ ⊕ D󸀠 . Hence, Cn ⊆ Cn−1 and Cn−1 /Cn is finite, which implies that C0 /Cn is finite. Finally, we have r0 (A0 /An ) = r0 (An−1 /An ) + r0 (A0 /An−1 ) ≤ n, as desired. This completes the induction.

Exercises. 1. (Megibben–Ullery [117]) The set of all k-subgroups of a group G is closed under ascending unions. 2. (Megibben–Ullery [117]) Let A be a subgroup of G such that |A| ≤ κ. If C is a k-subgroup of G containing A, then there is a k-subgroup B of G such that A ⊆ B ⊆ C and |B| ≤ κ.

3.12 Knice subgroups and k-groups Let G be a group. Then a nice subgroup A of G is called knice if for every finite subset S of G there is a finite (possibly empty) set of primitive elements x1 , . . . , xn ∈ G such

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that B = A ⊕ ⟨x1 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨xn ⟩ is a ∗-valuated coproduct in G with ⟨S, B⟩/B finite. If 0 is a knice subgroup of G, then G is called a k-group. Clearly, we have: (a) A nice subgroup A of G is knice exactly if for every finite subset S of G there is a finite ∗-decomposition set X in G such that A ⊕ ⟨X⟩ is a ∗-valuated coproduct in G containing a nonzero multiple of ⟨S⟩. (b) If Ai are knice subgroups of groups Gi (i ∈ I), then ⨁i∈I Ai is knice in ⨁i∈I Gi . Theorem 3.12.1 (Hill–Megibben [50]). Finite extensions of knice subgroups are knice. Proof. Suppose that A and B are subgroups of G such that A ⊆ B, B/A is finite, and A is knice in G. Then B is nice in G by Theorem 3.8.10. If S is a finite subset of G, then there is a ∗-valuated coproduct A ⊕ ⟨x1 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨xn ⟩ in G containing a nonzero multiple of ⟨S⟩ such that x1 , . . . , xn are primitive in G. By Proposition 3.6.5, there are nonzero multiples xi󸀠 of xi (i = 1, . . . , n) such that C = B⊕⟨x1󸀠 ⟩⊕⋅ ⋅ ⋅⊕⟨xn󸀠 ⟩ is a ∗-valuated coproduct in G. Since x1󸀠 , . . . , xn󸀠 are primitive in G, and ⟨S, C⟩/C is finite, it follows that B is knice in G. Proposition 3.12.2 (Hill–Megibben [50]). Suppose that A and B are subgroups of G such that A ⊆ B. If A is knice in G and B/A is nice in G/A, then B is nice in G. Proof. Let p be a prime, α an ordinal, and x + B ∈ pα (G/B). Assuming that px + B ∈ (pα G + B)/B, we have x + b + A ∈ pα (G/A) for some b ∈ B by Proposition 3.8.9(1). Since A is knice in G, there is a c ∈ G such that A ⊕ ⟨c⟩ is a valuated coproduct in G and n(x + b) = a + c for some positive integer n and a ∈ A. Then by Proposition 3.8.15, we have ≥ α; |c|Gp = |c + A|G/A p hence, n(x+b) = a+c ∈ pα G+A. We write n = pr n󸀠 , where p ∤ n󸀠 . Since x+b+A ∈ pα (G/A) and A is nice in G, this yields n󸀠 (x + b) ∈ pα G + A; thus, n󸀠 x ∈ pα G + B. Then there are integers k and l such that kn󸀠 +lp = 1; therefore, x = kn󸀠 x +lpx ∈ pα G +B. Consequently, B is nice in G. Theorem 3.12.3 (Hill–Megibben [50]). Suppose that B = A ⊕ ⟨x1 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨xn ⟩ is a ∗-valuated coproduct in a group G such that A is knice in G, and each element xi is primitive. Then B is knice in G. Proof. By induction, it suffices to show the assertion for n = 1. Let S be a finite subset of G. Then there are primitive elements y1 , . . . , ym in G such that C = A ⊕ ⟨y1 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨ym ⟩ is a ∗-valuated coproduct in G containing k⟨S, x1 ⟩ for some positive integer k. Since x1󸀠 = kx1 ∈ C, Theorem 3.10.14 shows that there is a ∗-valuated coproduct C 󸀠 = A ⊕ ⟨x1󸀠 ⟩ ⊕ ⟨x2 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨xm ⟩ in G with x2 , . . . , xm primitive in G, C 󸀠 ⊆ C, and C/C 󸀠 finite. Now B/(A ⊕ ⟨x1󸀠 ⟩) is finite; therefore, by Proposition 3.6.5 there are nonzero multiples xi󸀠 of 󸀠 xi (i = 2, . . . , m) such that B ⊕ ⟨x2󸀠 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨xm ⟩ is a ∗-valuated coproduct in G. Clearly, ⟨S, B⟩/B is finite. Since B is nice in G (Theorem 3.8.12), it follows that B is knice in G.

164 | 3 Warfield groups Corollary 3.12.4 (Hill–Megibben [50]). Let A be a knice subgroup of G. If S is a finite subset of G, then there is a knice subgroup C of G containing A and S such that C/A is finitely generated. In particular, if G is a k-group, then every finite subset is contained in some finitely generated knice subgroup of G. Proof. By the definition of knice and Theorem 3.12.3, there is a knice subgroup B of G containing A such that B/A is finitely generated and ⟨S, B⟩/B is finite. Now let C = ⟨S, B⟩, and apply Theorem 3.12.1. The next result follows from the proof of [53, Proposition 1.7]: Lemma 3.12.5. Suppose that A is quasi-sequentially nice in a group G, M is an Ulm matrix and p a prime. Then we have: 1. If x + A ∈ (G/A)(M ∗ , p), then nx ∈ G((nM)∗ , p) + A for some positive integer n. 2. If x + A ∈ (G/A)(M ∗ ), then nx ∈ G((nM)∗ ) + A for some positive integer n. 3. If x + A ∈ (G/A)(α∗ , p), then nx ∈ G((pr α)∗ , p) + A for some positive integer n = pr n󸀠 with p ∤ n󸀠 . Proof. To prove (1), suppose that x + A ∈ (G/A)(M ∗ , p). Then x + A ∈ (G/A)(M), and we can write x + A = z1 + ⋅ ⋅ ⋅ + zm + A, where zi + A ∈ (G/A)(M ∗ , p)⬦ for all i = 1, . . . , m. Then there is a positive integer n and a, a1 , . . . , am ∈ A such that ‖nx + a‖G = ‖nx + A‖G/A , and for i = 1, . . . , m, ‖nzi + ai ‖G = ‖nzi + A‖G/A . Let x󸀠 = nx + a and xi = nzi + ai . Notice that 󵄩󵄩 󸀠 󵄩󵄩 󵄩 󵄩 󸀠 󵄩󵄩x 󵄩󵄩G = 󵄩󵄩󵄩x + A󵄩󵄩󵄩G/A = ‖nx + A‖G/A ≥ nM by Proposition 3.8.15. Let i ∈ {1, . . . , m}. If zi + A ∈ (G/A)(M ∗ )⬦ , then—as before— we have ‖xi ‖G = ‖xi + A‖G/A . Hence, ‖xi ‖G ≥ nM and ‖xi ‖G ≁ nM. Similarly, if zi + A ∈ (G/A)(Mp∗ , p)⬦ , then we have ‖xi ‖Gp = ‖xi + A‖G/A = ‖nzi + A‖G/A ≥ (nM)p and |pj xi |Gp = p p |pj xi +A|G/A = |pj nzi +A|G/A ≠ mp,j+|n|p for infinitely many j < ω; thus, xi ∈ G((nM)∗p , p)⬦ . p p

Consequently, we have x1 , . . . , xm ∈ G((nM)∗ , p)⬦ . Since nx + A = nz1 + ⋅ ⋅ ⋅ + nzm + A = x1 + ⋅ ⋅ ⋅ + xn + A ⊆ G((nM)∗ , p) + A, we have nx ∈ G((nM)∗ , p) + A, as desired. (2) and (3) follow from the proof of (1). The proof of [53, Proposition 1.7] shows the following result: Lemma 3.12.6. Let A be a quasi-sequentially nice subgroup of a group G and X a ∗-decomposition set in G such that A ⊕ ⟨X⟩ is a ∗-valuated coproduct in G. Then {x + A : x ∈ X} is a ∗-decomposition set in G/A. Proof. Let x ∈ X, and notice that ‖x‖G = ‖x + A‖G/A by Proposition 3.8.15. Let M be an Ulm matrix and p a prime. To prove that x + A is primitive in G/A, assume that kx + A ∈ (G/A)(M ∗ , p) for some positive integer k. Then by Lemma 3.12.5(2), there is a positive integer n such that nkx ∈ G((nM)∗ , p)+A. By assumption, A⊕⟨x⟩ is a ∗-valuated coproduct in G; thus, nkx ∈ G((nM)∗ , p). Since x is primitive in G, we have ‖x + A‖G/A =

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‖x‖G ≁ nM ∼ M, or else nkx ∈ G((nM)∗p , p)⬦ ; that is, ‖nkx + A‖G/A = ‖nkx‖Gp ≥ (nM)p , p

and |pi nkx + A|G/A = |pi nkx|Gp ≠ mp,i+|n|p for infinitely many i < ω. It follows that p

‖x + A‖G/A ≁ M, or else kx + A ∈ (G/A)((nM)∗p , p)⬦ ; hence, x + A is primitive in G/A. It remains to show that ⨁x∈X ⟨x + A⟩ is a ∗-valuated coproduct. Note that it is at least a valuated coproduct in G/A by Corollary 3.8.16. Let z = r1 x1 + ⋅ ⋅ ⋅ + rm xm , where 0 ≠ ri ∈ ℤ, xi ∈ X (i = 1, . . . , m), and let i ∈ {1, . . . , m}. First, we assume that z +A ∈ (G/A)(M ∗ , p) and use Lemma 3.12.5(2) again to obtain nz ∈ G((nM)∗ , p)+A for some positive integer n. Since A⊕⟨x1 ⟩⊕⋅ ⋅ ⋅⊕⟨xm ⟩ is a ∗-valuated coproduct in G, we have nri xi ∈ G((nM)∗ , p); thus, nri xi + A ∈ (G/A)((nM)∗ , p) by Lemma 3.6.2(2). Since ri xi +A is primitive in G/A, we have ‖ri xi +A‖G/A ≁ nM ∼ M, or else ‖nri xi +A‖G/A ≥ (nM)p and |pj nri xi +A|Gp ≠ mp,j+|n|p for infinitely many j < ω. On the other p hand, we have ri xi +A ∈ (G/A)(M) since z +A ∈ (G/A)(M) and ⨁x∈X ⟨x +A⟩ is a valuated coproduct in G/A. It follows that ri xi + A ∈ (G/A)(M) ∩ (G/A)(M ∗ , p)⬦ ⊆ (G/A)(M ∗ , p), as desired. Next we assume that z + A ∈ (G/A)(M ∗ ). By Lemma 3.12.5(3), we have nz ∈ G((nM)∗ ) + A for some positive integer n. Then nri xi ∈ G((nM)∗ ); hence, nri xi + A ∈ (G/A)((nM)∗ ) (Lemma 3.6.2(2)). By Proposition 3.10.3(1) we obtain ‖nri xi + A‖G/A ≁ nM. Since ri xi + A ∈ (G/A)(M), this yields ri xi + A ∈ (G/A)(M ∗ ). Finally, suppose that z + A ∈ (G/A)(α∗ , p). Then Lemma 3.12.5(4) shows that nx ∈ G((pr α)∗ , p) + A for some positive integer n = pr n󸀠 with p ∤ n󸀠 . It follows that nri xi ∈ G((pr α)∗ , p), and therefore, nri xi + A ∈ (G/A)((pr α)∗ , p) (Lemma 3.6.2(2)). By Proposition 3.10.3(2), we have nri xi + A ∈ (G/A)((pr α)∗ , p)⬦ . Then ri xi + A ∈ (G/A)(α, p) shows that ri xi + A ∈ (G/A)(α∗ , p). This completes the proof. Knice subgroups are characterized as follows: Proposition 3.12.7 (Hill–Megibben [53]). Let A be a subgroup of a group G. Then the following conditions are equivalent: 1. A is knice in G; 2. A is quasi-sequentially nice in G, and G/A is a k-group. Proof. Suppose that A is knice in G. Let x ∈ G. Then there is a finite ∗-decomposition set X such that A ⊕ ⟨X⟩ is a ∗-valuated coproduct containing nx for some nonzero integer n. Thus, for some a ∈ A, nx + a ∈ ⟨X⟩ and A ⊕ ⟨nx + a⟩ is a valuated coproduct. Then by Lemma 3.9.6(3), A is quasi-sequentially nice. To prove that G/A is a k-group, let S be a finite subset of G. Since A is knice, there is a finite ∗-decomposition set X in G such that A ⊕ ⟨X⟩ is a ∗-valuated coproduct in G containing a nonzero multiple of ⟨S⟩. Then {x + A : x ∈ X} is a ∗-decomposition set in G/A by Lemma 3.12.6; hence, G/A is a k-group. To prove the converse, assume that A is quasi-sequentially nice in G, and that G/A is a k-group. Let S be a finite subset of G. Then there is a finite ∗-decomposition set {x1 + A, . . . , xn + A} in G/A such that ⟨x1 + A⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨xn + A⟩ contains a nonzero

166 | 3 Warfield groups multiple of ⟨S, A⟩/A. By Corollary 3.10.8 with B = A, there are positive integers mi and elements xi󸀠 ∈ mi xi + A (i = 1, . . . , n) such that C = A ⊕ ⟨x1󸀠 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨xn󸀠 ⟩ is a ∗-valuated coproduct in G and x1󸀠 , . . . , xn󸀠 are primitive in G. Since ⟨S, C⟩/C is finite, it follows that A is knice in G. Corollary 3.12.8 (Megibben–Ullery [116]). Suppose that B is a pure knice subgroup of a group G. If A is a subgroup of B, then B/A is a pure knice subgroup of G/A. Proof. By Proposition 3.12.7, B is quasi-sequentially nice in G and G/B is a k-group. Then Corollary 3.9.5 shows that B/A is pure and quasi-sequentially nice in G/A. Since (G/A)/(B/A) ≅ G/B is a k-group, B/A is knice in G/A by Proposition 3.12.7. Theorem 3.12.9 (Hill–Megibben [50]). Let A and B be subgroups of a group G such that A ⊆ B. If A is knice in G and B/A is knice in G/A, then B is knice in G. Proof. Let S be a finite subset of G. Then there is a ∗-valuated coproduct B/A⊕⟨x1 +A⟩⊕ ⋅ ⋅ ⋅ ⊕ ⟨xn + A⟩ in G/A containing m(⟨S, A⟩/A) for some positive integer m, where each element xi + A is primitive in G/A. Let i ∈ {1, . . . , n}. By Proposition 3.12.7, A is quasisequentially nice in G. Therefore, by Corollary 3.10.8, there is a positive integer mi such that xi󸀠 = mi xi is primitive in G, and C = B ⊕ ⟨x1󸀠 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨xn󸀠 ⟩ is a ∗-valuated coproduct in G. Then ⟨S, C⟩/C is finite. Since B is a nice subgroup of G by Proposition 3.12.2, it follows that B is knice in G. The following result was extracted from the proof of [116, Theorem 2.1]: Proposition 3.12.10. Let A ⊕ B be a ∗-valuated coproduct in a group G, and let F = ⟨X⟩, where X is a countable ∗-decomposition set in G. Suppose that S is a finite subset of A with ⟨S, F⟩/F finite, and that F/((F ∩ A) ⊕ (F ∩ B)) is torsion. Then there is a finite ∗-decomposition set Y in G such that Y ⊆ A and ⟨S, Y⟩/⟨Y⟩ is finite. Proof. Since ⟨S, F⟩/F is finite, there exists a finite subset X0 = {x1 , . . . , xn } of X such that ⟨S, F0 ⟩/F0 is finite, where F0 = ⟨X0 ⟩. Then, by Corollary 3.11.8, there are subgroups A󸀠 and B󸀠 of F and nonzero multiples xi󸀠 of xi (i = 1, . . . , n) such that 1. F 󸀠 = ⟨x1󸀠 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨xn󸀠 ⟩ ⊕ A󸀠 ⊕ B󸀠 is a ∗-valuated coproduct in G; 2. F/F 󸀠 is torsion; 3. A󸀠 ⊆ A and (F ∩ A)/A󸀠 has torsion-free rank ≤ n. Let F0󸀠 = ⟨x1󸀠 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨xn󸀠 ⟩, and notice that ⟨S, F0󸀠 ⟩/F0󸀠 is finite since F0 /F0󸀠 and ⟨S, F0 ⟩/F0 are finite. Letting A󸀠󸀠 = A ∩ (F0󸀠 ⊕ B󸀠 ), we have F 󸀠 ∩ A = (F0󸀠 ⊕ A󸀠 ⊕ B󸀠 ) ∩ A = A󸀠 ⊕ (A ∩ (F0󸀠 ⊕ B󸀠 )) = A󸀠 ⊕ A󸀠󸀠 . By Corollary 3.11.1 and Proposition 3.11.2(1), F 󸀠 is k-basic in G since F/F 󸀠 is torsion, and A󸀠 ⊕A󸀠󸀠 ⊕(F 󸀠 ∩B) = (F 󸀠 ∩A)⊕(F 󸀠 ∩B) is k-basic in G since F 󸀠 /((F 󸀠 ∩A)⊕(F 󸀠 ∩B)) is torsion. Now A󸀠󸀠 ≅ (F 󸀠 ∩ A)/A󸀠 has torsion-free rank ≤ n, and A󸀠 ⊕ A󸀠󸀠 ⊕ (F 󸀠 ∩ B) is a ∗-valuated

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coproduct in G. Therefore, by Corollary 3.11.6, there exists a finite ∗-decomposition set Y in G such that Y ⊆ A󸀠󸀠 and A󸀠󸀠 /⟨Y⟩ is torsion. Since ⟨S, A󸀠󸀠 ⟩/A󸀠󸀠 ≅ ⟨S⟩/(⟨S⟩ ∩ A󸀠󸀠 ) = ⟨S⟩/(⟨S⟩ ∩ (F0󸀠 ⊕ B󸀠 )) ≅ ⟨S, F0󸀠 ⊕ B󸀠 ⟩/F0󸀠 ⊕ B󸀠 is finite, we conclude that the finitely generated group ⟨S, Y⟩/⟨Y⟩ is torsion, hence finite. Since Y ⊆ A, the proof is complete. The class of k-groups is closed under direct summands: Theorem 3.12.11 (Hill–Megibben [53]). A direct summand of a k-group is a k-group. Proof. Suppose that G = H ⊕ K is a k-group, and let S be a finite subset of H. Then there is a ∗-decomposition set T0 = {x1 , . . . , xn0 } in G such that ⟨S, F0 ⟩/F0 is finite, where F0 = ⟨T0 ⟩. We write xi = hi + ki (hi ∈ H, ki ∈ K) for i = 1, . . . , n0 , and let S0 = {h1 , . . . , hn0 , k1 , . . . , kn0 }. By Theorem 3.12.3, F0 is knice in G. Therefore, T0 is contained in some ∗-decomposition set T1 = {x1 , . . . , xn0 , xn0 +1 , . . . , xn1 } in G such that ⟨S0 , F1 ⟩/F1 is finite, where F1 = ⟨T1 ⟩. Write xi = hi + ki (hi ∈ H, ki ∈ K) for i = n0 + 1, . . . , n1 , and let S1 = {h1 , . . . , hn1 , k1 , . . . , kn1 }. Again, we can find a ∗-decomposition set T2 = {x1 , . . . , xn2 } in G containing T1 such that ⟨S1 , F2 ⟩/F2 is finite with F2 = ⟨T2 ⟩, et cetera. This gives us ascending sequences {Tn }n σ. Now let z be any element of S. Then |x − y󸀠 + z|p = |(x + z) − y󸀠 |p = |x + z|p ≤ |x|p = σ. Thus, x − y󸀠 is proper with respect to S. This gives us an element, x − y󸀠 ∈ G[p] of height σ proper with respect to S. By Corollary 4.5.5, we have dim Dσ,p (S) < up (σ, G), and since S is finitely generated, Dσ,p (S) is of finite dimension. Thus, dim Dσ,p (S) < û p (σ, G) ≤ û p (σ, H). Therefore, by Lemma 4.5.3, we obtain dim Dσ,p (T) < û p (σ, H). By Corollary 4.5.5, there is a w1 ∈ H, proper with respect to T, such that |w1 |p = σ and pw1 = 0. Since f is height-preserving up to α > σ + 1, |f (px)|p > σ + 1, so we may choose a w2 ∈ H such that pw2 = f (px) and |w2 |p > σ. Let y = w1 + w2 . Then py = pw2 = f (px), and |y|p = |w1 |p ∧ |w2 |p = σ. We claim that y is proper with respect to T. Let z ∈ T. Then |y + z|p = |(w1 + z) + w2 |p = |w1 + z|p ≤ σ. This completes Case 2. In either case, we have a y that is proper with respect to T, with |y|p = σ, and py = f (px). In particular, y ∈ ̸ T since |y| ≠ ∞. Now we extend f by sending x to y, giving us the isomorphism f ∗ : ⟨S, x⟩ → ⟨T, y⟩ (Lemma 1.12.4(2)). We need only prove that f ∗ preserves p-heights up to α. Let s + ax be an arbitrary element of ⟨S, x⟩, where s ∈ S and a ∈ R. If p|a, then s + ax ∈ S. If not, we may write 1 = bp + ra for some b, r ∈ R. Then r(s + ax) = rs − bpx + x ∈ S + x. Thus, every z ∈ ⟨S, x⟩ is either in S, or else rz ∈ S + x for some r ∈ R not divisible by p. In the first case, we are done since f ∗ (z) = f (z). So suppose rz = s + x for some s ∈ S and r not divisible by p. Then |z|p = |rz|p = |s + x|p = |s|p ∧ |x|p since x is proper with respect to S, and similarly, 󵄨 󵄨 󵄨󵄨 ∗ 󵄨󵄨 󵄨󵄨f (z)󵄨󵄨p = 󵄨󵄨󵄨f (s)󵄨󵄨󵄨p ∧ |y|p = |s|p ∧ |x|p since y is proper with respect to T. The following version of this theorem is a key extension theorem for proving that two groups are partially isomorphic.

210 | 5 Groups with partial decomposition bases Theorem 5.1.10. Let G and H be R-modules, p a prime in R, and f an isomorphism between a finitely generated submodule S of G and a submodule T of H that preserves p-heights. Suppose that û p (σ, G) ≤ û p (σ, H) for all ordinals σ. If x ∈ G is proper with respect to S, px ∈ S, and |x|p ≠ ∞, then for a suitable y in H, f can be extended to an isomorphism f ∗ : ⟨S, x⟩ → ⟨T, y⟩ that preserves p-heights. Proof. Let α > max{p-length(G), p-length(H)} + 1, and apply Theorem 5.1.9. In the next sections, we will be proving classification theorems both up to partial isomorphism and up to ≡λ for certain ordinals λ. We will prove in the various cases that, if the appropriate modified Ulm and Warfield invariants agree for PDB groups G and H, then we get extension properties that allow us to apply the following theorem: Theorem 5.1.11. Let G and H be R-modules with rich partial decomposition bases 𝒞 and 𝒟, respectively, where R = ℤ or R = ℤp for some prime p. Let λ be an ordinal, and suppose that the following conditions hold for any height-preserving (or heightpreserving up to ω(ν + 1) for some ν < λ) isomorphism f : S → T, where S ⊆ G and T ⊆ H are finitely generated submodules: (a) If X ⊆ S ⊆ ⟨X⟩0 for some X in 𝒞 , where f (X) ∈ 𝒟, and if X ∪ {x} ∈ 𝒞 for some x ∈ G, then there are a nonzero multiple x 󸀠 of x and a y󸀠 ∈ H such that f (X) ∪ {y󸀠 } ∈ 𝒟 and g : ⟨S, x󸀠 ⟩ → ⟨T, y󸀠 ⟩, extending f by x 󸀠 󳨃→ y󸀠 , is an isomorphism that preserves heights (or preserves heights up to ω(ν + 1)); (b) If a ∈ G has a nonzero multiple in S, then there is b ∈ H such that g : ⟨S, a⟩ → ⟨T, b⟩, extending f by a 󳨃→ b, is an isomorphism that preserves heights (or preserves heights up to ων); and analogous conditions on partial isomorphisms from H to G. Then G ≅p H (or G ≡λ H). Specifically, suppose X0 ∈ 𝒞 , Y0 ∈ 𝒟, possibly empty, and f0 : X0 → Y0 is a bijection such that ‖x‖ = ‖f0 (x)‖ for all x ∈ X0 . Let I (or Iν with ν ≤ λ) be the set of all maps f : S → T with associated X ∈ 𝒞 and Y ∈ 𝒟 such that (i) S and T are finitely generated submodules of G and H, respectively; (ii) f is a height-preserving (or ων-height-preserving) isomorphism; (iii) X ⊆ S ⊆ ⟨X⟩0 , and Y ⊆ T ⊆ ⟨Y⟩0 ; (iv) f (X) = Y; (v) X0 ⊆ X, and f ↾ X0 = f0 . Then I (or {Iν : ν ≤ λ}) is a Karp system. Proof. This system is not empty, since it contains f0 . Note that if μ ≤ ν, then Iν ⊆ Iμ . We will prove that this system has the back-and-forth property. Let f : S → T be in I (or Iν+1 ) with associated X and Y, and let a ∈ G. Then, since 𝒞 is a partial decomposition basis, there is some X 󸀠 ∈ 𝒞 such that X ⊆ X 󸀠 and a ∈ ⟨X 󸀠 ⟩0 , say X 󸀠 = X ∪ {x1 , . . . , xn }. We will define, by induction on 0 ≤ i ≤ n, suitable

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maps fĩ : S̃i → T̃ i in I (or Iν+1 ) with associated X̃ i = X ∪ {x1󸀠 , . . . , xi󸀠 } and Ỹ i , where S̃i = ⟨S, x1󸀠 , . . . , xi󸀠 ⟩, and each xi󸀠 is a nonzero multiple of xi . Start the induction with f0̃ = f , X̃ 0 = X, S̃0 = S and T̃ 0 = T. Now suppose we have defined fĩ , S̃i , T̃ i , X̃ i , and Ỹ i for 0 ≤ i < n. By (a), there is a height-preserving (or ω(ν + 1)-height-preserving) ̃ : S̃ = ⟨S̃ , x󸀠 ⟩ → ⟨T̃ , y󸀠 ⟩, for some x󸀠 a multiple of x and y󸀠 isomorphism fi+1 i+1 i i+1 i i+1 i+1 i+1 i+1 󸀠 󸀠 󸀠 󸀠 such that Ỹ i ∪ {yi+1 } ∈ 𝒟. Let Ỹ i+1 = Ỹ i ∪ {yi+1 }, X̃ i+1 = X̃ i ∪ {xi+1 } and T̃ i+1 = ⟨T̃ i , yi+1 ⟩. It is ̃ satisfies all of (i) through (v) and so is in I (or I ). This completes then clear that fi+1 ν+1 the induction and gives us fñ ∈ I (or Iν+1 ) with associated S̃n , T̃ n , X̃ n , and Ỹ n , where X̃ n = X ∪ {x1󸀠 , . . . , xn󸀠 }. Since a has a nonzero multiple in ⟨X 󸀠 ⟩, it has one in ⟨X̃ n ⟩ ⊆ S̃n . So by (b), there is a b ∈ H such that g : ⟨S̃n , a⟩ → ⟨T̃ n , b⟩, extending fn , is an isomorphism that preserves heights (or preserves heights up to ων). Then the map g with associated X̃ n ∈ 𝒞 and Ỹ n ∈ 𝒞 satisfies all of (i) through (v) and so is in I (or Iν ). Also, a ∈ domain(g), as desired. The other direction follows by symmetry.

Exercises. 1. Prove that 𝒞n in the proof of Lemma 5.1.3 is a partial decomposition basis for every n. 2. Construct a partial decomposition basis for ∏ω ℤ. (Hint: Use the proof of Theorem 4.1.4.)

5.2 Classification of local groups up to partial isomorphism In this section, we focus on p-local PDB groups for some fixed prime p. This means we may use any of the previous results that apply to groups, ℤp -modules, or arbitrary R-modules. We wish to construct an invariant analogous to the Warfield invariant for PDB modules, one that is maintained under ≅p . Recall that Barwise and Eklof used the modified Ulm invariant û p (α, G) = up (α, G) ∧ ω to classify torsion groups up to partial isomorphism. We may do the same thing with the invariant s(e, G) and define a modified Stanton invariant: For e an equivalence class of Ulm sequences and G a ℤp -module, we define ̂ G) = s(e, G) ∧ ω. s(e, We cannot do the same for w(e, G) since the definition relies on the existence of a decomposition basis, but we can define the modified Warfield invariant as follows: For e an equivalence class of Ulm sequences and G a ℤp -module with partial decomposition basis 𝒞 , let ŵ 𝒞 (e, G) = the maximum n < ω such that there are X ∈ 𝒞 and x1 , . . . , xn ∈ X with ‖xi ‖p ∈ e for 1 ≤ i ≤ n, if such a maximum exists, and ω otherwise.

212 | 5 Groups with partial decomposition bases Theorem 5.2.1 (Jacoby [68], Jacoby–Loth [74]). Let G be a ℤp -module with partial decomposition bases 𝒞 and 𝒞 󸀠 . Suppose for some equivalence class e of Ulm sequences ŵ 𝒞 (e, G) ≥ n. Then for any Y ∈ 𝒞 󸀠 , there is a Ỹ ∈ 𝒞 󸀠 such that Y ⊆ Ỹ and Ỹ contains at least n elements y such that ‖y‖p ∈ e. Proof. By the definition of w,̂ there is an X ∈ 𝒞 containing at least n elements with Ulm sequences in e. Choose X 󸀠 and Y 󸀠 as in Theorem 5.1.4, building on X and Y. Now consider the submodule ⟨X 󸀠 ⟩0 = ⟨Y 󸀠 ⟩0 of G as a ℤp -module. Heights are the same in this module as they are in the whole of G; therefore, X 󸀠 is a decomposition basis, as is Y 󸀠 . Since X 󸀠 contains at least n elements with Ulm sequence in e, we have w(e, ⟨X 󸀠 ⟩0 ) ≥ n. By Theorem 3.2.3, the Warfield invariant is independent of the decomposition basis, so the decomposition basis Y 󸀠 must contain n such elements as well. Since Y 󸀠 was built as an ascending chain of elements of 𝒞 󸀠 , we may choose some Ỹ in this chain that contains these n elements. Corollary 5.2.2. Let G be a ℤp -module with partial decomposition bases 𝒞 and 𝒞 󸀠 . Then for any equivalence class e of Ulm sequences, ŵ 𝒞 (e, G) = ŵ 𝒞 󸀠 (e, G). Proof. Suppose ŵ 𝒞 (e, G) ≥ n. Then, using any Y ∈ 𝒞 󸀠 , the theorem says that there is some Ỹ ∈ 𝒞 󸀠 with at least n elements with Ulm sequence in e. It follows that ŵ 𝒞 󸀠 (e, G) ≥ n, and so ŵ 𝒞 (e, G) ≤ ŵ 𝒞 󸀠 (e, G). The other direction follows by symmetry. The corollary shows that ŵ 𝒞 (e, G) is indeed an invariant, in the sense that it is independent of the choice of partial decomposition basis. Therefore, we will drop the subscript 𝒞 in what follows. We may now restate Theorem 5.2.1. Theorem 5.2.3. Let G be a ℤp -module with partial decomposition basis 𝒞 . Suppose for ̂ G) ≥ n. Then for any X ∈ 𝒞 , there is an some equivalence class e of Ulm sequences w(e, X̃ ∈ 𝒞 such that X ⊆ X̃ and X̃ contains at least n elements x with ‖x‖p ∈ e. ̂ G) is invariant under partial isomorphism, as we can prove by Furthermore, w(e, an appropriate choice of partial decomposition basis: Corollary 5.2.4. Let G and H be ℤp -modules such that G has a partial decomposition ̂ H) is basis 𝒞 and I : G ≅p H. Then for any equivalence class e of Ulm sequences, w(e, ̂ ̂ defined, and we have w(e, H) = w(e, G). Proof. Let 𝒞 󸀠 = {f (X) : X ∈ 𝒞 , f ∈ I, X ⊆ domain(f )}. Theorem 5.1.2 tells us that 𝒞 󸀠 is a ̂ H) partial decomposition basis for H. Since H has a partial decomposition basis, w(e, is defined for any equivalence class of Ulm sequences e. Now suppose for some e, ̂ G) ≥ n. Then by the theorem, there is some X ∈ 𝒞 and some x1 , . . . , xn ∈ X with w(e, ‖xi ‖p ∈ e for all 1 ≤ i ≤ n. Since X is finite, we may choose an f ∈ I with X ⊆ domain(f ). Since f is height-preserving, ‖f (xi )‖p ∈ e for i = 1, . . . , n. Then f (X) ∈ 𝒞 󸀠 includes n ̂ H) ≥ n. Conversely, suppose elements with Ulm sequence equivalent to e, and so w(e, ̂ H) ≥ n. Then there are f (x1 ), . . . , f (xn ) ∈ f (X) for some f ∈ I and X ∈ 𝒞 with the w(e,

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property that for each i = 1, . . . , n, ‖f (xi )‖p ∈ e. Then for each 1 ≤ i ≤ n, xi ∈ X and ‖xi ‖p ∈ e. We will need two more lemmas before we can prove the local classification theorem. Lemma 5.2.5 (Jacoby [68]). Let G be a reduced ℤp -module of finite torsion-free rank with a decomposition basis {x1 , . . . , xn }. Then there is a torsion ℤp -module T and submodules Gi of G (i = 1, . . . , n) such that G ⊕ T ≅p G1 ⊕ ⋅ ⋅ ⋅ ⊕ Gn , and xi ∈ Gi for all i = 1, . . . , n. In fact, the set I of all height-preserving isomorphisms between finitely generated submodules of G⊕T and G1 ⊕⋅ ⋅ ⋅⊕Gn , extending the canonical map φ : ⟨x1 , . . . , xn ⟩ → ⟨x1 , . . . , xn ⟩, satisfies the conditions of Karp’s Theorem 4.3.2. Proof. For each 1 ≤ i ≤ n, let Gi = ⟨xi ⟩0 ⊆ G, and let T be the direct sum of n−1 copies of the torsion part of G. Let S and L be the respective submodules of G ⊕ T and G1 ⊕ ⋅ ⋅ ⋅ ⊕ Gn generated by {x1 , . . . , xn }. Let φ : S → L be the isomorphism such that φ(xi ) = xi for i = 1, . . . , n. Note that {x1 , . . . , xn } is a decomposition basis for G ⊕ T and for G1 ⊕ ⋅ ⋅ ⋅ ⊕ Gn and that the height of any multiple of any xi is the same in G, G ⊕ T, or G1 ⊕ ⋅ ⋅ ⋅ ⊕ Gn . It follows that φ is height-preserving. Now we will construct a suitable Karp system. Let I be the set of all f : A → B such that (i) A ⊆ G ⊕ T and B ⊆ G1 ⊕ ⋅ ⋅ ⋅ ⊕ Gn are finitely generated submodules, (ii) S ⊆ A, L ⊆ B, (iii) f is a height-preserving isomorphism and (iv) φ ⊆ f . Since φ ∈ I, I is not empty. We need only prove that I has the back-and-forth property. Let f : A → B be a map in I, and z + w ∈ G ⊕ T, where z ∈ G and w ∈ T. For some m ≥ 0, pm (z + w) ∈ S since G ⊕ T is torsion over S. We define by induction on k, 0 ≤ k ≤ m, maps fk : Ak → Bk in I such that f ⊆ fk , pm−k (z + w) ∈ Ak , and Ak /A is finite. Let f0 = f . Suppose fk has been defined. Then, since S is p-nice in G ⊕ T by Lemma 5.1.5 and Ak /S is finitely generated and torsion, Ak is p-nice as well (Lemma 2.1.4(2)). We may then choose x proper in pm−(k+1) (z + w) + Ak . Note that px ∈ Ak . If x ∈ S, let fk+1 = fk . Suppose x ∈ ̸ S. Notice that G⊕T and G1 ⊕⋅ ⋅ ⋅⊕Gn have the same Ulm invariants since their torsion parts agree. By Theorem 5.1.10, we may extend fk to an isomorphism fk+1 : ⟨Ak , x⟩ → ⟨Bk , y⟩ that preserves heights. Let Ak+1 = ⟨Ak , x⟩, and Bk+1 = ⟨Bk , y⟩. Then it may be verified that fk+1 : Ak+1 → Bk+1 is in I, pm−(k+1) (z + w) ∈ x + Ak ⊆ Ak+1 , and Ak+1 /Ak is finite. This completes the induction. Finally, let g = fm . Then z + w ∈ Am = domain(g), g ∈ I, and f ⊆ g, as desired.

214 | 5 Groups with partial decomposition bases Similarly, if z1 + ⋅ ⋅ ⋅ + zn ∈ G1 ⊕ ⋅ ⋅ ⋅ ⊕ Gn with zi ∈ Gi (i = 1, . . . , n), then for some m ≥ 0, pm zi ∈ ⟨xi ⟩ for all 1 ≤ i ≤ n. Then pm (z1 + ⋅ ⋅ ⋅ + zn ) ∈ ⟨x1 , . . . , xn ⟩, so we may extend f −1 to ⟨B, z1 + ⋅ ⋅ ⋅ + zn ⟩ as in the other case. This proves that I has the back-and-forth property, and so the proof is complete. Lemma 5.2.6 (Jacoby–Loth [72]). Let G be a ℤp -module, X a decomposition basis for G, and S a finitely generated submodule of G such that S ∩ ⟨X⟩ = ⟨S ∩ X⟩. If y ∈ X such that y ∉ S then there is an n < ω such that 󵄨 󵄨 n 󵄨 󵄨󵄨 n 󵄨󵄨rp y + s󵄨󵄨󵄨 = 󵄨󵄨󵄨rp y󵄨󵄨󵄨 ∧ |s| for all r ∈ ℤp and s ∈ S. Proof. Since S ∩ ⟨y⟩ ⊆ ⟨S ∩ X⟩ and y ∈ ̸ S, S ∩ ⟨y⟩ = 0. We may write S ∩ X = {x1 , . . . , xm } since S is finitely generated. We may assume that the first k elements are exactly the ones with Ulm sequence not equivalent to (∞, ∞, . . . ), for some 0 ≤ k ≤ m. We may assume ‖y‖p ≁ (∞, ∞, . . . ), since otherwise |pn y| = ∞ for some n, and the claim follows. Now let H = (S ⊕ ⟨y⟩)0 , and let D be the maximal divisible subgroup of H. Then {y, x1 , . . . , xm } is a decomposition basis for H. Also, ⟨y, x1 , . . . , xk ⟩ ∩ D = 0. Therefore, by Theorem 1.6.3, there is a subgroup R of H such that ⟨y, x1 , . . . , xk ⟩ ⊆ R and H = R ⊕ D. Since R is a reduced ℤp -module with decomposition basis {y, x1 , . . . , xk }, we may apply Lemma 5.2.5 to get I : R ⊕ T ≅p H0 ⊕ ⋅ ⋅ ⋅ ⊕ Hk , where T is torsion, y ∈ H0 , and xi ∈ Hi for i = 1, . . . , k. For any f : A → B in I, we define f 󸀠 : A ⊕ D → B ⊕ D by (a, x) 󳨃→ (f (a), x). Then I 󸀠 = {f 󸀠 : f ∈ I} is a Karp system between R⊕T ⊕D ≅ H ⊕T and H0 ⊕⋅ ⋅ ⋅⊕Hk ⊕D. Note that f 󸀠 (x) = x for all f 󸀠 ∈ I 󸀠 and x ∈ {y, x1 , . . . , xm }. Then the finitely generated subgroup S ⊕ ⟨y⟩ is in the domain of some f 󸀠 ∈ I 󸀠 . Now let π be the projection of H0 ⊕ ⋅ ⋅ ⋅ ⊕ Hk ⊕ D onto H0 . We claim πf 󸀠 (S) is torsion. Let z ∈ S. Then for some r ∈ ℤp , rz ∈ S ∩ ⟨X⟩ = ⟨S ∩ X⟩. Then rπf 󸀠 (z) = πf 󸀠 (rz) = πrz = 0, proving the claim. Since S is finitely generated, πf 󸀠 (S) is as well. This says that πf 󸀠 (S) takes on only finitely many heights. But since ‖y‖p ≁ (∞, ∞, . . . ) and f 󸀠 is height-preserving, each pn f 󸀠 (y) takes on a different height. So there is some n < ω such that |pn f 󸀠 (y)| > |s0 | for all s0 ∈ πf 󸀠 (S). In fact, |rpn f 󸀠 (y)| > |s0 | for all r ∈ ℤp \ {0}, and s0 ∈ πf 󸀠 (S). Now let s ∈ S and r ∈ ℤp . Write f 󸀠 (s) = s0 + ⋅ ⋅ ⋅ + sk + d, where si ∈ Hi for 0 ≤ i ≤ k and d ∈ D. Then |rpn y + s| = |f 󸀠 (rpn y + s)| = |f 󸀠 (rpn y) + s0 | ∧ |s1 | ∧ ⋅ ⋅ ⋅ ∧ |sk | ∧ |d| = |f 󸀠 (rpn y)| ∧ |f 󸀠 (s)| = |rpn y| ∧ |s|. Now we are ready for the local classification theorem for PDB groups. Theorem 5.2.7 (Jacoby [68]). Let G and H be PDB modules. Then G ≅p H if and only ̂ G) = u(α, ̂ H) and if for every α ∈ Ord∞ and equivalence class e of Ulm sequences, u(α, ̂ G) = w(e, ̂ H). w(e, In that case, if 𝒞 and 𝒞 󸀠 are rich partial decomposition bases of G and H, respectively, and if X0 ∈ 𝒞 , Y0 ∈ 𝒞 󸀠 , and f0 : X0 → Y0 is a bijection such that ‖x‖p = ‖f0 (x)‖p for all x ∈ X0 , then I : G ≅p H may be chosen to be the set of all maps f : S → T, for which there exist X ∈ 𝒞 and Y ∈ 𝒞 󸀠 , satisfying the following properties:

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(i) S and T are finitely generated submodules of G and H, respectively; (ii) f is a height-preserving isomorphism; (iii) X ⊆ S ⊆ ⟨X⟩0 , and Y ⊆ T ⊆ ⟨Y⟩0 ; (iv) f (X) = Y; (v) X0 ⊆ X, and f ↾ X0 = f0 . Proof. First, suppose G ≅p H. Then all û and ŵ invariants agree by Corollaries 4.4.6 and 5.2.4. Now suppose that G and H have the same û and ŵ invariants. Let 𝒞 and 𝒞 󸀠 be respective rich partial decomposition bases for G and H. By Theorem 5.1.11, it suffices to prove that the following conditions hold for any height-preserving isomorphism f : S → T, where S ⊆ G and T ⊆ H are finitely generated submodules: (a) If X ⊆ S ⊆ ⟨X⟩0 for some X in 𝒞 , where f (X) ∈ 𝒞 󸀠 , and if X ∪ {x} ∈ 𝒞 for some x ∈ G, then there is a nonzero multiple x󸀠 of x and a y󸀠 ∈ H such that f (X) ∪ {y󸀠 } ∈ 𝒞 󸀠 and g : ⟨S, x󸀠 ⟩ → ⟨T, y󸀠 ⟩, extending f by x󸀠 󳨃→ y󸀠 , is an isomorphism that preserves heights; (b) If a ∈ G has a nonzero multiple in S, then there is a b ∈ H such that g : ⟨S, a⟩ → ⟨T, b⟩, extending f by a 󳨃→ b, is an isomorphism that preserves heights; and analogous conditions on functions from H to G. Case (a): We may assume x ∈ ̸ S, since otherwise the proof is trivial; hence, x ∉ X. Suppose X 󸀠 = X ∪ {x} ∈ 𝒞 . Let Y = f (X), and let e be the equivalence class of ‖x‖p . ̂ ⟨X⟩0 ) = |{x ∈ X : ‖x‖p ∈ e}| = w(e, ̂ ⟨Y⟩0 ). Since f is height-preserving, we have w(e, ̂ ⟨Y⟩0 ) = w(e, ̂ ⟨X⟩0 ) < w(e, ̂ ⟨X ∪ {x}⟩0 ) ≤ w(e, ̂ G) = w(e, ̂ H). By Theorem 5.2.3, Then w(e, 󸀠 󸀠 󸀠 there is a Y ∈ 𝒞 where Y ⊆ Y , and a y ∈ Y 󸀠 \ Y with ‖y‖p ∼ ‖x‖p . Since 𝒞 󸀠 is rich, we may assume Y 󸀠 = Y ∪ {y}. Now apply Lemma 5.2.6 to ⟨S, x⟩0 , the submodule S, the decomposition basis X 󸀠 , and x ∈ X 󸀠 . This applies since S ∩ ⟨X 󸀠 ⟩ = ⟨X⟩ = ⟨S ∩ X 󸀠 ⟩, and x ∈ ̸ S. Apply it similarly to ⟨T, y⟩0 , T, Y 󸀠 , and y. This gives us an n < ω such that for all s ∈ S, t ∈ T, and r ∈ ℤp , |rpn x + s| = |rpn x| ∧ |s| and |rpn y + t| = |rpn y| ∧ |t|. Since ‖pn x‖p ∼ ‖pn y‖p , there are x󸀠 and y󸀠 , nonzero multiples of pn x and pn y, such that ‖x󸀠 ‖p = ‖y󸀠 ‖p . Define g : ⟨S, x󸀠 ⟩ → ⟨T, y󸀠 ⟩, where g ↾ S = f and g(x󸀠 ) = y󸀠 . By the choice of n, g is height-preserving. This completes Case (a). Case (b): It suffices to consider the case of a ∈ G \ S and pa ∈ S. By Lemma 5.1.5, ⟨X⟩ is p-nice in G. Since ⟨X⟩ ⊆ S ⊆ ⟨X⟩0 , S/⟨X⟩ is torsion and finitely generated, hence finite. It follows by Lemma 2.1.4(2) that S is p-nice, so we may assume a is proper with respect to S. If |a| ≠ ∞, we may extend f to a by Theorem 5.1.10, specifically for some b ∈ H, there is a height-preserving g : ⟨S, a⟩ → ⟨T, b⟩ with f ⊆ g. Suppose |a| = ∞. Then |f (pa)| = ∞, so f (pa) = pb󸀠 for some b󸀠 ∈ H with |b󸀠 | = ∞. 󸀠 If b ∈ ̸ T, we may extend f to an isomorphism by a 󳨃→ b󸀠 by Lemma 1.6.5. So suppose b󸀠 ∈ T, say b󸀠 = f (z) for some z ∈ S. Then f (pa − pz) = 0, and so pa − pz = 0, which means a − z ∈ (p∞ G)[p]. Since a − z ∈ ̸ S, dim(p∞ H ∩ T)[p] = dim(p∞ G ∩ S)[p] < ̂ ̂ dim(p∞ G ∩ ⟨S, a − z⟩)[p] ≤ u(∞, G) = u(∞, H). It follows that there exists an element

216 | 5 Groups with partial decomposition bases b̃ ∈ (p∞ H)[p] \ (p∞ H ∩ T)[p]. In particular, b̃ ∈ ̸ T, |b|̃ = ∞, and pb̃ = 0. Let b = b󸀠 + b.̃ Then pb = f (pa), b ∈ ̸ T, and |b| = ∞, as desired. We may now extend f by a 󳨃→ b, and the extended function is a height-preserving isomorphism. The specific structure of I comes from the construction in the proof of Theorem 5.1.11. Note that, in the theorem above, we can always choose f0 = 0, and X0 = Y0 = 0, making (v) trivial.

5.3 Classification of general groups up to partial isomorphism The classification of local groups serves as a stepping-stone to classifying general, global groups. A key piece to get us there is Warfield’s local-global lemma (see Lemma 3.4.5): Let G and H be groups and A and B subgroups of G and H, respectively, such that G/A and H/B are torsion. Suppose f : A → B is a homomorphism such that for every prime p, the induced homomorphism fp : Ap → Bp extends to a homomorphism g(p) : Gp → Hp . Then f extends to a homomorphism g : G → H such that gp = g(p) for all primes p. If every map g(p) is an isomorphism, then g is an isomorphism. We will use modified global Warfield and Stanton invariants based on those developed for the isomorphism case. Recall the global Warfield and Stanton invariants from Section 3.7: For a compatibility class c of Ulm matrices, prime p, and equivalence class e of Ulm sequences, the (global) Stanton invariant of a group G is defined by s(c, p, e, G) = sup{rp,0 (G(M)/G(M ∗ , p)) : M ∈ c and Mp ∈ e}, and if G has a decomposition basis X, the (global) Warfield invariant of G is given by 󵄨 󵄨 wX (c, p, e, G) = 󵄨󵄨󵄨{x ∈ X : ‖x‖ ∈ c and ‖x‖p ∈ e}󵄨󵄨󵄨. Recall that if G has a decomposition basis X, then the two invariants coincide for all c, p, e (Theorem 3.7.3), and so wX (c, p, e, G) is independent of the choice of decomposition basis. This allows us to write simply w(c, p, e, G) without ambiguity. We define the modified Warfield invariant for a group G with partial decomposition basis 𝒞 , compatibility class c, prime p, and equivalence class e as ŵ 𝒞 (c, p, e, G) = the largest n < ω if it exists, such that there are X ∈ 𝒞 and x1 , . . . , xn ∈ X with ‖xi ‖ ∈ c and ‖xi ‖p ∈ e for 1 ≤ i ≤ n. If no such n exists, we let ŵ 𝒞 (c, p, e, G) = ω. Note that if G has a decomposition basis X and 𝒞 consists of all finite subsets of X, ̂ p, e, G) = s(c, p, e, G) ∧ ω this cardinal is equal to the modified Stanton invariant, s(c,

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for all c, p, and e. Later we will see that this holds for any group with a partial decomposition basis. As in the local case, this invariant is independent of the choice of 𝒞 , so we may drop the subscript 𝒞 : Theorem 5.3.1. If a group G has partial decomposition bases 𝒞 and 𝒞 󸀠 , then for any compatibility class c, prime p, and equivalence class e of Ulm sequences, ŵ 𝒞 (c, p, e, G) = ŵ 𝒞 󸀠 (c, p, e, G). In fact, if ŵ 𝒞 (c, p, e, G) ≥ n and Y ∈ 𝒞 󸀠 , there is a Ỹ ∈ 𝒞 󸀠 such that Y ⊆ Ỹ and Ỹ contains distinct elements y1 , . . . , yn such that ‖yi ‖ ∈ c and ‖yi ‖p ∈ e for i = 1, . . . , n. Proof. Suppose ŵ 𝒞 (c, p, e, G) ≥ n for some c, p, and e, say x1 , . . . , xn ∈ X, where X ∈ 𝒞 , and for all 1 ≤ i ≤ n, ‖xi ‖ ∈ c, and ‖xi ‖p ∈ e. Construct X 󸀠 and Y 󸀠 based on this X and the given Y as in Theorem 5.1.4. Then consider ⟨X 󸀠 ⟩0 = ⟨Y 󸀠 ⟩0 as a group with heights the same as in G. Both X 󸀠 and Y 󸀠 are decomposition bases, so by Theorem 3.7.3, w(c, p, e, ⟨Y 󸀠 ⟩0 ) = w(c, p, e, ⟨X 󸀠 ⟩0 ) ≥ n. Thus, Y 󸀠 contains the desired y1 , . . . , yn . Since Y 󸀠 is the union of an ascending chain of elements of 𝒞 󸀠 , we may choose Ỹ ⊆ Y 󸀠 such that Ỹ ∈ 𝒞 󸀠 , Y ⊆ Y,̃ and y1 , . . . , yn ∈ Y.̃ As we would expect, the modified Warfield invariant is in fact invariant under ≅p . ̂ p, e, G) = w(c, ̂ p, e, H) Lemma 5.3.2. Let G and H be PDB groups. If I : G ≅p H, then w(c, for each compatibility class c of Ulm matrices, prime p, and equivalence class e of Ulm sequences. Proof. Let 𝒞 be a partial decomposition basis for G. Then {f (X) : X ∈ 𝒞 , f ∈ I} is a ̂ p, e, G) ≥ n. Then there partial decomposition basis for H (Theorem 5.1.2). Suppose w(c, is some X ∈ 𝒞 containing at least n elements with ‖x‖ ∈ c and ‖x‖p ∈ e. Choose f ∈ I such that X ⊆ domain(f ). Then f (X) contains at least n elements y with ‖y‖ ∈ c and ̂ p, e, G) ≤ w(c, ̂ p, e, H) and the other direction follows by ‖y‖p ∈ e. This proves w(c, symmetry. Lemma 5.3.3. Let G and H be groups with partial decomposition bases 𝒞 and 𝒞 󸀠 , respectively. Let c be a compatibility class of Ulm matrices, p a prime, and e an equivâ p, e, G) = w(c, ̂ p, e, H), X ∈ 𝒞 , Y ∈ 𝒞 󸀠 , and lence class of Ulm sequences. Suppose w(c, ̂ p, e, ⟨X⟩0 ) > w(c, ̂ p, e, ⟨Y⟩0 ). Then there is a Y 󸀠 ∈ 𝒞 󸀠 such that Y ⊆ Y 󸀠 , and a y ∈ Y 󸀠 \Y w(c, such that ‖y‖ ∈ c and ‖y‖p ∈ e. ̂ p, e, ⟨Y⟩0 ), we get w(c, ̂ p, e, H) = w(c, ̂ p, e, G) ≥ w(c, ̂ p, e, ⟨X⟩0 ) > Proof. Letting n = w(c, ̂ p, e, ⟨Y⟩0 ) = n. Thus, w(c, ̂ p, e, H) ≥ n + 1. By Theorem 5.3.1, there is a Y 󸀠 ∈ 𝒞 󸀠 such w(c, 󸀠 󸀠 that Y ⊆ Y and Y contains at least n + 1 elements y with ‖y‖ ∈ c and ‖y‖p ∈ e. Since Y contains only n such elements, one must be in Y 󸀠 \ Y. Lemma 5.3.4. Suppose G is a PDB group, p is a prime, and e is an equivalence class of ̂ p, e, G)) ∧ ω. ̂ Gp ) = (∑c w(c, Ulm sequences. Then w(e,

218 | 5 Groups with partial decomposition bases Proof. Let 𝒞 be a partial decomposition basis for G. The image of 𝒞 under the p-heightpreserving natural map G → Gp is a partial decomposition basis for Gp ; hence, ̂ p, e, G) ≥ n, say ̂ p, e, G) ≥ n. Suppose that ∑c w(c, ̂ Gp ) ≥ n implies ∑c w(c, w(e, k ̂ i , p, e, G) ≥ n for some 1 ≤ k ≤ n. By induction, define X1 ⊆ ⋅ ⋅ ⋅ ⊆ Xk , where each ∑i=1 w(c ̂ i , p, e, G) elements x such that ‖x‖ ∈ ci and ‖x‖p ∈ e. Xi ∈ 𝒞 and contains at least w(c This can be done by Theorem 5.3.1. The natural map G → Gp preserves heights, so the image of Xk contains at least n elements x with ‖x‖p ∈ e, completing the proof. Lemma 5.3.3 allows us to find an additional element in H associated with given c, p and e as long as the Warfield invariant is large enough. To prove the classification theorem, we will need to find such an element that matches the c, p, and e for a given x ∈ G for every c, p, and e. The following variation on a result of Stanton will help us get there. It allows us to modify a decomposition basis one prime at a time, giving us the extension lemma that follows it. Lemma 5.3.5 (Stanton [140]). Let G be a group with a rich partial decomposition basis 𝒞 . Suppose X ∪ {x1 , x2 } ∈ 𝒞 , where x1 and x2 have compatible Ulm matrices, and let p be a prime. Then there are elements y1 and y2 in G such that ‖y1 ‖p = ‖x2 ‖p , ‖y2 ‖p = ‖x1 ‖p , and ‖y1 ‖q = ‖x1 ‖q , ‖y2 ‖q = ‖x2 ‖q for all primes q ≠ p. Moreover, if Y = X ∪ {y1 , y2 }, then Y ∈ 𝒞 and ⟨X ∪ {x1 , x2 }⟩ = ⟨Y⟩. Proof. Since X ∪ {x1 , x2 } is a decomposition set, ⟨x1 ⟩ ⊕ ⟨x2 ⟩ is a valuated coproduct in G. By Proposition 3.7.4, we then get y1 and y2 with the desired Ulm sequences. Also, ⟨X ∪ {x1 , x2 }⟩ = ⟨X⟩ ⊕ ⟨x1 ⟩ ⊕ ⟨x2 ⟩ = ⟨X⟩ ⊕ ⟨y1 ⟩ ⊕ ⟨y2 ⟩ = ⟨Y⟩. In particular, Y ∈ 𝒞 since 𝒞 is rich. Lemma 5.3.6. Let G and H be groups with rich partial decomposition bases 𝒞 and 𝒟, ̂ p, e, G) = w(c, ̂ p, e, H) for every compatibility class c of Ulm respectively. Suppose w(c, matrices, prime p, and equivalence class e of Ulm sequences. Then if X ∪ {x} ∈ 𝒞 , Y ∈ 𝒟, ̂ p, e, ⟨X⟩0 ) = w(c, ̂ p, e, ⟨Y⟩0 ) for all c, p, and e, then there is a y ∈ H such that and w(c, ̂ p, e, ⟨X ∪ {x}⟩0 ) = w(c, ̂ p, e, ⟨Y ∪ {y}⟩0 ) for all c, p, and e. In fact, ‖x‖ Y ∪ {y} ∈ 𝒟 and w(c, and ‖y‖ are equivalent Ulm matrices; that is, k‖x‖ = l‖y‖ for some positive integers k, l. Proof. We start with a fixed prime p0 . Let c be the compatibility class of ‖x‖ and e the equivalence class of ‖x‖p0 . Choose a z ∈ H \ Y as in Lemma 5.3.3. Specifically, ‖z‖ ∈ c, ‖z‖p0 ∈ e, and Y ∪ {z} ∈ 𝒟. Then ‖z‖ and ‖x‖ are compatible. In particular, ‖z‖p and ‖x‖p are equivalent for all but finitely many primes p1 , . . . , pn . By induction on n, we will find our desired y, where Y ∪ {y} ∈ 𝒟, ‖y‖ ∈ c, and ‖y‖p ∼ ‖x‖p for all p. Since ‖y‖p = ‖x‖p for almost all p, it will follow that ‖y‖ and ‖x‖ are equivalent. If n = 0, z works as the desired y. Suppose the claim is true for n − 1. Let en = [‖x‖pn ]. We know by assumption on pn that ‖z‖pn ≁ ‖x‖pn , so ‖z‖pn ∈ ̸ en . It follows ̂ pn , en , ⟨Y ∪ {z}⟩0 ) = w(c, ̂ pn , en , ⟨Y⟩0 ) < w(c, ̂ pn , en , ⟨X ∪ {x}⟩0 ). Choose a z 󸀠 ∈ that w(c, H as in Lemma 5.3.3 applied to Y ∪ {z} and X ∪ {x}. Then Y ∪ {z, z 󸀠 } ∈ 𝒟, ‖z 󸀠 ‖ ∈ c, and ‖z 󸀠 ‖pn ∈ en . Now we use Lemma 5.3.5 to swap {z, z 󸀠 } for some {y, y󸀠 }. Specifically,

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⟨Y ∪ {y, y󸀠 }⟩ = ⟨Y ∪ {z, z 󸀠 }⟩, ‖y‖pn = ‖z 󸀠 ‖pn , and ‖y‖q = ‖z‖q for all q ≠ pn . Then ‖y‖pn = ‖z 󸀠 ‖pn ∼ ‖x‖pn , and ‖y‖q = ‖z‖q ∼ ‖x‖q for all q ∈ ̸ {p1 , . . . , pn }. In particular, ‖y‖p ∼ ‖x‖p for all p ∈ ̸ {p1 , . . . , pn−1 }. The result follows by the induction hypothesis. Lemma 5.3.7. Let G be a group, X a decomposition basis for G, and S a finitely generated subgroup of G such that S ∩ ⟨X⟩ = ⟨S ∩ X⟩. Then if y ∈ X and y ∉ S, there is a positive integer n such that |mny + s|p = |mny|p ∧ |s|p for all m ∈ ℤ, s ∈ S, and primes p. Proof. This lemma was proved in the local case, Lemma 5.2.6. We will localize to use that result. Since S is finitely generated and S ⊆ ⟨X⟩0 , we may find an integer k ≠ 0 such that ks ∈ ⟨X⟩ for all s ∈ S. Let p be a prime and νp : G → Gp the natural map. We claim νp (y) ∈ ̸ S ⊗ ℤp . Suppose to the contrary that m(y ⊗ 1) = s󸀠 ⊗ 1 for some m ∈ ℤ \ {0} and s󸀠 ∈ S. Then my − s󸀠 is torsion (Lemma 1.17.8(3)); hence, m󸀠 y ∈ S for some m󸀠 ∈ ℤ \ {0}. Then m󸀠 y ∈ S ∩ ⟨X⟩ = ⟨S ∩ X⟩, which means that y ∈ S ∩ X, contradicting the original assumption that y ∈ ̸ S, completing the claim. Now apply Lemma 5.2.6 to Gp with decomposition basis νp (X), and νp (y) ∈ ̸ S ⊗ ℤp . There is an np ∈ ℤ such that |rpnp νp (y) + νp (s)|p = |rpnp νp (y)|p ∧ |νp (s)|p for all r ∈ ℤp and s ∈ S. Now let n = Πp|k pnp . We will prove that this is our desired n. First suppose p divides k, and hence, pnp divides n. Then for any m ∈ ℤ and s ∈ S, we have |mny + s|p = |mnνp (y) + νp (s)|p = |mnνp (y)|p ∧ |νp (s)|p . Next suppose p does not divide k, and let s ∈ S. Since ks ∈ S ∩ ⟨X⟩ = ⟨S ∩ X⟩, we may write ks = a1 x1 + ⋅ ⋅ ⋅ + al xl for some x1 , . . . , xl ∈ S ∩ X. In particular, none of x1 , . . . , xl a is y. Then since p does not divide k, we may write νp (s) = ak1 νp (x1 ) + ⋅ ⋅ ⋅ + kn νp (xn ), and so νp (s) ∈ ⟨νp ({x1 , . . . , xn })⟩. Finally, since νp ({x1 , . . . , xn , y}) is a decomposition set, |mny + s|p = |νp (mny) + νp (s)|p = |νp (mny)|p ∧ |νp (s)|p = |mny|p ∧ |s|p , as required. Now we are ready for the global classification theorem. Theorem 5.3.8. Let G and H be PDB groups. Then G ≅p H if and only if for every α ∈ Ord∞ , prime p, compatibility class c of Ulm matrices, and equivalence class e of Ulm ̂ p, e, G) = w(c, ̂ p, e, H). sequences, û p (α, G) = û p (α, H) and w(c, Proof. First suppose that G ≅p H. Then by Corollary 4.4.6 and Lemma 5.3.2, the modified Ulm and Warfield invariants agree. Now suppose the modified Ulm and Warfield invariants agree. We may choose rich partial decomposition bases 𝒞 and 𝒞 󸀠 for G and H, respectively. By Theorem 5.1.11, we need only prove that the following conditions hold for any height-preserving isomorphism f : S → T, where S ⊆ G and T ⊆ H are finitely generated subgroups: (a) If X ⊆ S ⊆ ⟨X⟩0 for some X in 𝒞 , where f (X) ∈ 𝒞 󸀠 , and if X ∪ {x} ∈ 𝒞 for some x ∈ G, then there is a nonzero multiple x󸀠 of x and a y󸀠 ∈ H such that f (X) ∪ {y󸀠 } ∈ 𝒞 󸀠 and g : ⟨S, x󸀠 ⟩ → ⟨T, y󸀠 ⟩, extending f by x󸀠 󳨃→ y󸀠 , is an isomorphism that preserves heights;

220 | 5 Groups with partial decomposition bases (b) If a ∈ G has a nonzero multiple in S, then there is a b ∈ H such that g : ⟨S, a⟩ → ⟨T, b⟩, extending f , is an isomorphism that preserves heights; and analogous conditions on functions from H to G. Case (a): Suppose X ∪ {x} ∈ 𝒞 . We will extend f to some multiple of x. Let Y = f (X). By Lemma 5.3.6, there is a y ∈ H \ Y such that Y ∪ {y} ∈ 𝒞 󸀠 and ‖x‖ and ‖y‖ are equivalent. Then there are multiples x󸀠 and y󸀠 of x and y, respectively, such that ‖x󸀠 ‖ = ‖y󸀠 ‖. Consider the group ⟨S, x󸀠 ⟩0 with decomposition basis X 󸀠 = X ∪ {x󸀠 }. Since S ∩ ⟨X 󸀠 ⟩ = ⟨X⟩ = ⟨S ∩ X 󸀠 ⟩, we may apply Lemma 5.3.7 to X 󸀠 and S, and similarly to Y 󸀠 = Y ∪{y󸀠 } and T. That gives us a positive integer n such that |mnx󸀠 +s|p = |mnx󸀠 |p ∧|s|p and |mny󸀠 + t|p = |mny󸀠 |p ∧ |t|p for all m ∈ ℤ, p prime, s ∈ S, and t ∈ T. Now define g : ⟨S, nx 󸀠 ⟩ → ⟨T, ny󸀠 ⟩, extending f by nx 󸀠 󳨃→ ny󸀠 . Case (b): Suppose a has a nonzero multiple in S. Let p be a prime. Then the localization map νp maps 𝒞 and 𝒞 󸀠 to partial decomposition bases of Gp and Hp , respectively. ̂ p, e, H)) ∧ ω = w(e, ̂ Hp ). Also ̂ p, e, G)) ∧ ω = (∑c w(c, ̂ Gp ) = (∑c w(c, By Lemma 5.3.4, w(e, ̂ Gp ) = u(α, ̂ Hp ) for all α ∈ Ord∞ by Lemma 1.17.11. u(α, Now we may apply the local classification theorem 5.2.7 to get Ip : Gp ≅p Hp . Since Ip is a Karp system, there is a function in Ip extending fp with νp (a) in its domain. Let G󸀠 = ⟨S, a⟩, H 󸀠 = T 0 . Let g(p) be that function restricted to Gp󸀠 . Now apply Lemma 3.4.5, the local-global lemma, to the groups G󸀠 and H 󸀠 , torsion over their respective subgroups S and T, and f : S → T. By that lemma, f extends to an injective homomorphism g : ⟨S, a⟩ → T 0 such that gp = g(p) for all primes p. Thus, g is an isomorphism onto its range. Finally, g is height-preserving since each g(p) is p-heightpreserving. Exercises. 1. (Göbel–Leistner–Loth–Strüngmann [37]) Give a counterexample to disprove the following statement: Let f : S → T be a height-preserving isomorphism, where S and T are submodules of some ℤp -modules G and H. Then if x ∈ G and y ∈ H such that for some n > 0, pn x ∈ S, pn y ∈ T, pn−1 x ∈ ̸ S, pn−1 y ∉ T, and f (pn x) = pn y, then f can be extended to a height-preserving isomorphism ⟨S, x⟩ → ⟨T, y⟩ by x 󳨃→ y.

5.4 Classification up to an ordinal Now that we have completed the classification up to partial isomorphism, we return to the question of equivalence up to some ordinal. Recall from the previous chapter that one version of Karp’s Theorem 4.3.1 gives us two different ways of looking at such an equivalence: 1. 2.

Let G and H be groups and λ ∈ Ord∞ . Then the following are equivalent: G ≡λ H; For each ordinal ν ≤ λ, there is a nonempty set Iν of isomorphisms on finitely generated subgroups of G into H such that

5.4 Classification up to an ordinal | 221

(a) if ν ≤ μ, then Iμ ⊆ Iν ; (b) if ν < λ, f ∈ Iν+1 , and x ∈ G (or y ∈ H), then f extends to a map f 󸀠 ∈ Iν such that x ∈ domain(f 󸀠 ) (or y ∈ range(f 󸀠 )). Recall that a system of sets Iν (ν ≤ λ) as in item (2) is called a Karp system. We will prove classification theorems in Lλ∞ω by constructing appropriate Karp systems. These will depend highly on the idea of “up to λ” for various ordinals. This viewpoint focuses on what happens below some threshold ordinal, and ignores differences, for example in heights, above the threshold. The following definitions will give the flavor of this approach and will be used throughout this section. Let α be an ordinal. Recall from Section 4.4 the following definitions: Two ordinals β and γ are called equal up to α, written β =α γ if β ∧ α = γ ∧ α. Then β ≥α γ is defined similarly. Given groups G and H, a map f : G S → T with S ⊆ G and T ⊆ H is α-height-preserving if and only if |f (x)|H p =α |x|p for all x ∈ S and primes p. For Ulm sequences and Ulm matrices, =α and ≥α are defined analogously. Two Ulm sequences u = (βi )i dim(S[p] ∩ p∞ G) = dim(S[p] ∩ p∞ (tG)) = dim(S[p] ∩ pα (tG)), since length(tG) < ων + k < α. Since f is α-height-preserving, this is dim(T[p] ∩ pα H). This proves that dim(p∞ H)[p] > dim(T[p] ∩ pα H). Therefore, there is an element in (p∞ H)[p] that is not in T. We may write it as pj b1 , where |b1 | = ∞. This is our desired b1 . We may now take b0 + b1 as our desired b. Case 2b: Now suppose length(tG) ≥ ω(ν + 1). Then length(tH) ≥ ω(ν + 1) as well, since otherwise the Ulm invariants would disagree between length(tH) and ω(ν + 1). Again, we seek a b as the image of x. By Lemma 4.5.1, applied to tT ⊆ tH, the limit ordinal ω(ν + 1) ≤ length(tH), and β = ων + m + k, we get b1 ∈ pων+m+k H such that pj b1 ∈ ̸ T and pj+1 b1 = 0. As before, take b = b0 + b1 . This completes Case 2b. In either case, we have produced a b ∈ H such that pj b ∉ T, pj+1 b = f (pj+1 x), and |b| ≥ ων + m + k. By Lemma 5.4.3, we may define f 󸀠 : ⟨S, x⟩ → ⟨T, b⟩, extending f by x 󳨃→ b, an ων + m + k-height-preserving isomorphism. Now we will extend f 󸀠 to include a. First, we claim that ⟨S, x⟩ = ⟨S, pm a⟩. Recall that x ∈ pm a + ⟨S, pm+1 a⟩, say x = pm a + s + rpm+1 a for some r ∈ ℤp and s ∈ S. Let y = pm a + rpm+1 a = (1 + rp)pm a. Then ⟨S, x⟩ = ⟨S, y⟩, since x − y ∈ S. Since p ∤ (1 + rp), ⟨y⟩ = ⟨pm a⟩, and so domain(f 󸀠 ) = S󸀠 = ⟨S, x⟩ = ⟨S, pm a⟩. Now let l ≤ m − 1, and z ∈ pl a + ⟨S󸀠 , pl+1 a⟩ = pl a + ⟨S, pl+1 a⟩. By the minimality of m, |z| < ω(ν + 1). Since z and l were arbitrary, we may apply Case 1 to f 󸀠 : S󸀠 → T 󸀠 , and m − 1. Thus, we may extend f 󸀠 to g, an ων + k-height-preserving isomorphism with domain ⟨S󸀠 , a⟩. Now we look at extending isomorphisms to elements of a partial decomposition basis. This theorem provides the foundation. Theorem 5.4.5. Let G be a ℤp -module with partial decomposition basis 𝒞 , α an ordinal, and n a positive integer. Suppose ŵ α (e, G) ≥ n, and X ∈ 𝒞 . Then there is an X 󸀠 ∈ 𝒞 such that X ⊆ X 󸀠 and X 󸀠 has ≥ n elements x with ‖x‖p ∼α e.

5.4 Classification up to an ordinal | 225

Proof. Choose a finite set {e1 , . . . , em } of distinct equivalence classes with ei ∼α e for i = 1, . . . , m, such that ∑m i=1 w(ei , G) ≥ n. If w(ei , G) ≥ ω for some i, the result follows from Theorem 5.2.1. So suppose for each 1 ≤ i ≤ m, w(ei , G) is finite, say w(ei , G) = ni . Now we will define by induction on 0 ≤ i ≤ m, Xi ∈ 𝒞 , having at least ni elements x with ‖x‖p ∈ ei , and such that X = X0 ⊆ X1 ⊆ ⋅ ⋅ ⋅ ⊆ Xm . Given Xi , Theorem 5.2.1 gives us Xi+1 ⊇ Xi with ≥ ni+1 elements x with ‖x‖p ∈ ei+1 . This completes the induction. Note that these elements are distinct since the ei are distinct. Let X 󸀠 = Xm . Now we are ready to prove the classification theorem. If the modified Ulm and Warfield invariants agree up to a certain level, depending on λ, then the groups are λ-equivalent. Theorem 5.4.6 (Jacoby–Loth [72]). Let G and H be PDB modules. Let λ be an ordinal such that ̂ G) = u(α, ̂ H) for all α < ωλ; 1. u(α, 2. ŵ ω(ν+1) (e, G) = ŵ ω(ν+1) (e, H) for all ν < λ and equivalence classes e of Ulm sequences; ̂ ̂ 3. if length(tG) < ωλ, then u(∞, G) = u(∞, H). Then G ≡λ H. Specifically, for any (possibly empty) X0 and Y0 in the respective partial decomposition bases 𝒞 and 𝒞 󸀠 , and for any bijection f0 : X0 → Y0 such that ‖x‖p = ‖f0 (x)‖p for all x ∈ X0 , there is a Karp system {Iν : ν ≤ λ} with each set Iν consisting of all maps f : S → T with associated X ∈ 𝒞 and Y ∈ 𝒞 󸀠 such that (i) S and T are finitely generated submodules of G and H, respectively; (ii) f is an ων-height-preserving isomorphism; (iii) X ⊆ S ⊆ ⟨X⟩0 , and Y ⊆ T ⊆ ⟨Y⟩0 ; (iv) f (X) = Y; (v) X0 ⊆ X and f ↾ X0 = f0 . Proof. We may choose rich partial decomposition bases 𝒞 and 𝒞 󸀠 . By Theorem 5.1.11, it suffices to prove that the following conditions hold for any ordinal ν < λ and any ω(ν + 1)-height-preserving isomorphism f : S → T, where S ⊆ G and T ⊆ H are finitely generated submodules: (a) If X ⊆ S ⊆ ⟨X⟩0 for some X in 𝒞 , where f (X) ∈ 𝒞 󸀠 , and if X ∪ {x} ∈ 𝒞 for some x, then there is a nonzero multiple x󸀠 of x and a y󸀠 ∈ H such that f (X) ∪ {y󸀠 } ∈ 𝒞 󸀠 and g : ⟨S, x󸀠 ⟩ → ⟨T, y󸀠 ⟩, extending f by x󸀠 󳨃→ y󸀠 , is an isomorphism that preserves heights up to ω(ν + 1); (b) If a ∈ G has a nonzero multiple in S, then there is a b ∈ H such that g : ⟨S, a⟩ → ⟨T, b⟩, extending f by a 󳨃→ b, is an isomorphism that preserves heights up to ων; and analogous conditions on partial isomorphisms from H to G. Now let f : S → T be such a mapping.

226 | 5 Groups with partial decomposition bases Case (a): Let X and x be as in (a), and e = [‖x‖]. Then ŵ ω(ν+1) (e, H) = ŵ ω(ν+1) (e, G) ≥ ŵ ω(ν+1) (e, ⟨X ∪ {x}⟩0 ) > ŵ ω(ν+1) (e, ⟨X⟩0 ) = ŵ ω(ν+1) (e, ⟨Y⟩0 ), where Y = f (X). By Theorem 5.4.5, there is a Y 󸀠 ∈ 𝒞 󸀠 such that Y ⊆ Y 󸀠 and 0

ŵ ω(ν+1) (e, ⟨Y 󸀠 ⟩ ) ≥ ŵ ω(ν+1) (e, ⟨Y⟩0 ) + 1. Therefore, there must be a y ∈ Y 󸀠 \Y such that ‖y‖p ∼ω(ν+1) e, and hence, ‖y‖p ∼ω(ν+1)

‖x‖p . Then there are n󸀠 , m󸀠 < ω such that ‖pn x‖p =ω(ν+1) ‖pm y‖p . Assume x ∈ ̸ S, since the other case is trivial. Now apply Lemma 5.2.6 to the ℤp -module ⟨S, x⟩0 with decomposition basis X ∪ {x} and submodule S. This applies since X ∪ {x} ⊆ ⟨S, x⟩, and so, since S ⊆ ⟨X⟩0 , 󸀠

󸀠

S ∩ ⟨X ∪ {x}⟩ = ⟨X⟩ = ⟨S ∩ (X ∪ {x})⟩. We may similarly apply Lemma 5.2.6 to ⟨T, y⟩0 , Y ∪ {y}, and T, since T ⊆ ⟨Y⟩0 says y ∈ ̸ T. This gives us k < ω such that for all s ∈ S, t ∈ T, and r ∈ ℤp , 󵄨 󵄨 k 󵄨 󵄨󵄨 k 󵄨󵄨rp x + s󵄨󵄨󵄨 = 󵄨󵄨󵄨rp x󵄨󵄨󵄨 ∧ |s| and 󵄨 󵄨 k 󵄨 󵄨󵄨 k 󵄨󵄨rp y + t 󵄨󵄨󵄨 = 󵄨󵄨󵄨rp y󵄨󵄨󵄨 ∧ |t|. Now let n = k + n󸀠 , m = k + m󸀠 , and define g extending f by sending pn x to pm y. It is straightforward to verify that g is an isomorphism that preserves heights to ω(ν + 1). Case (b): Choose n ≥ 0 least such that pn+1 a ∈ S. Choose k < ω sufficiently large so that for any given j ≤ n, if every x ∈ pj a + ⟨S, pj+1 a⟩ has height < ω(ν + 1), then every such x has height < ων + k, and if length(tG) < ω(ν + 1) then length(tG) < ων + k. This is possible since the submodules involved are p-nice by Corollary 5.1.8, and so each coset has an element of maximal height. The map f preserves heights up to ω(ν + 1); hence, it preserves heights up to ων + k + n + 1. By Lemma 5.4.4, f extends to an ων + k-heightpreserving isomorphism g with domain(g) = ⟨S, a⟩. Thus, g is an ων-height-preserving isomorphism. This completes the classification of PDB modules. Now we consider arbitrary PDB groups. Recall that ŵ α (c, p, e, G) = (

∑ c󸀠 ∼α c,e󸀠 ∼α e

̂ 󸀠 , p, e󸀠 , G)) ∧ ω. w(c

̂ p, e, G) is the Recall also that if G has a partial decomposition basis 𝒞 , then w(c, largest n, if it exists, such that for some X ∈ 𝒞 and x1 , . . . , xn ∈ X, ‖xi ‖ ∈ c, and ‖xi ‖p ∈ e. ̂ p, e, ⟨X⟩0 ) = |{x ∈ X : ‖x‖ ∈ c, and ‖x‖p ∈ e}|. Hence, In particular, for any X, w(c, 0 ̂ p, e, G) ≥ w(c, ̂ p, e, ⟨X⟩ ). It then follows that ŵ α (c, p, e, G) ≥ ŵ α (c, p, e, ⟨X⟩0 ). w(c, As before, the local classification gives us the foundation for the global classification theorem. We previously used Lemma 5.3.5 to build the required system of partial isomorphisms. Here we prove the analogous lemma up to α.

5.4 Classification up to an ordinal | 227

Lemma 5.4.7. Let G be a group with decomposition basis X, p a prime, and x1 , x2 ∈ X with α-compatible Ulm matrices. Then there are elements y1 , y2 ∈ ⟨X⟩ such that ‖y1 ‖p =α ‖x2 ‖p , ‖y2 ‖p =α ‖x1 ‖p , and ‖y1 ‖q =α ‖x1 ‖q , ‖y2 ‖q =α ‖x2 ‖q for all primes q ≠ p. The set Y = (X\{x1 , x2 }) ∪ {y1 , y2 } is a decomposition basis for G, and ⟨X⟩ = ⟨Y⟩. Proof. Since ‖x1 ‖ ∼α ‖x2 ‖, we have some positive integer m such that m‖x1 ‖ ≥α ‖x2 ‖ and m‖x2 ‖ ≥α ‖x1 ‖. Write m = pj n, where p ∤ n. Now we follow the proof of Proposition 3.7.4. This gives us integers s and t such that sp2j − tn = 1 and proves that, if we define y spj ( 1) = ( y2 n

tn x1 )( ), pj x2

then ⟨x1 ⟩ ⊕ ⟨x2 ⟩ = ⟨y1 ⟩ ⊕ ⟨y2 ⟩ and ⟨y1 ⟩ ⊕ ⟨y2 ⟩ is a valuated coproduct. Then since ⟨X⟩ = ⟨X \ {x1 , x2 }⟩ ⊕ ⟨x1 ⟩ ⊕ ⟨x2 ⟩ = ⟨X \ {x1 , x2 }⟩ ⊕ ⟨y1 ⟩ ⊕ ⟨y2 ⟩ = ⟨Y⟩ is a valuated coproduct, Y is a decomposition set. Following the proof further, we get ‖y1 ‖p = ‖spj x1 ‖p ∧ ‖tnx2 ‖p =α ‖x2 ‖p , since ‖spj x1 ‖p = ‖smx1 ‖p ≥α ‖sx2 ‖p and ‖tnx2 ‖p = ‖x2 ‖p . Similarly, ‖y2 ‖p =α ‖x1 ‖p . Now suppose q ≠ p. Note that q does not divide both n and s, and so at least one of them is relatively prime to q. It may be verified that in either case ‖y1 ‖q =α ‖x1 ‖q , and ‖y2 ‖q =α ‖x2 ‖q . Theorem 5.4.8. Let G be a group with partial decomposition basis 𝒞 , α an ordinal, and n a positive integer. If ŵ α (c, p, e, G) ≥ n and Y ∈ 𝒞 , then there exists Y 󸀠 ∈ 𝒞 such that Y ⊆ Y 󸀠 and Y 󸀠 contains elements y1 , . . . , yn such that ‖yi ‖ ∼α c and ‖yi ‖p ∼α e for all i = 1, . . . , n. ̂ 󸀠 , p, e󸀠 , G) ≠ 0, c󸀠 ∼α c, and e󸀠 ∼α e}. Choose disProof. Let E = {(c󸀠 , e󸀠 ) : w(c tinct elements (c1 , e1 ), . . . , (ck , ek ) of E such that k ≤ n and ∑1≤i≤k mi ≥ n, where ̂ i , p, ei , G) for each 1 ≤ i ≤ k. This is possible since w(c ̂ i , p, ei , G) ≥ 1 for each mi = w(c 1 ≤ i ≤ k. We define Yi inductively for each 0 ≤ i ≤ k. Let Y0 = Y. Given Yi , by Theorem 5.3.1, there is a Yi+1 ∈ 𝒞 such that Yi ⊆ Yi+1 and Yi+1 contains at least mi+1 elements x such that ‖x‖ ∈ ci+1 ∼α c and ‖x‖p ∈ ei+1 ∼α e. Let Y 󸀠 = ⋃ki=0 Yi . Then Y 󸀠 ⊇ Y contains at least mi elements x such that ‖x‖ ∈ ci and ‖x‖p ∈ ei for each i, and such elements for mi and mj are distinct for i ≠ j. Hence, there are at least m1 + ⋅ ⋅ ⋅ + mk ≥ n elements in Y 󸀠 with ‖x‖ ∼α c and ‖x‖p ∼α e. Corollary 5.4.9. Let G be a group with partial decomposition basis 𝒞 , α an ordinal, c a compatibility class of Ulm matrices, p a prime, and e an equivalence class of Ulm sequences. Then ŵ α (c, p, e, G) is the largest integer n, if it exists, such that there are X ∈ 𝒞 and x1 , . . . , xn ∈ X with ‖xi ‖ ∼α c and ‖xi ‖p ∼α e for all i = 1, . . . , n. If no such n exists, ŵ α (c, p, e, G) = ω. Proof. First, suppose there is a largest n for which there is an X ∈ 𝒞 and x1 , . . . , xn ∈ X, such that ‖xi ‖ ∼α c and ‖xi ‖p ∼α e for all 1 ≤ i ≤ n. We know that ŵ α (c, p, e, G) ≥

228 | 5 Groups with partial decomposition bases ŵ α (c, p, e, ⟨X⟩0 ) = n. By the theorem, if ŵ α (c, p, e, G) > n, then there is some Y ∈ 𝒞 and y1 , . . . , yn+1 ∈ Y such that ‖yi ‖ ∼α c and ‖yi ‖p ∼α e for all 1 ≤ i ≤ n + 1, contradicting maximality. If there is no largest n, then for any n, there is an X ∈ 𝒞 and x1 , . . . , xn ∈ X with ‖xi ‖ ∼α c and ‖xi ‖p ∼α e for all 1 ≤ i ≤ n. Then ŵ α (c, p, e, G) ≥ ŵ α (c, p, e, ⟨X⟩0 ) ≥ n. Since n was arbitrary, ŵ α (c, p, e, G) = ω. Corollary 5.4.10. Let G and H be groups with partial decomposition bases 𝒞 and 𝒟, respectively, α an ordinal, c a compatibility class of Ulm matrices, p a prime, and e an equivalence class of Ulm sequences. Suppose ŵ α (c, p, e, G) = ŵ α (c, p, e, H), X ∈ 𝒞 , Y ∈ 𝒟, and ŵ α (c, p, e, ⟨X⟩0 ) > ŵ α (c, p, e, ⟨Y⟩0 ). Then there exists a Y 󸀠 ∈ 𝒟 such that Y ⊆ Y 󸀠 and there is an element y ∈ Y 󸀠 \Y such that ‖y‖ ∼α c and ‖y‖p ∼α e. Proof. Suppose ŵ α (c, p, e, ⟨Y⟩0 ) = n. Then ŵ α (c, p, e, G) ≥ ŵ α (c, p, e, ⟨X⟩0 ) ≥ n + 1. By the theorem, there is a Y 󸀠 ∈ 𝒞 such that Y ⊆ Y 󸀠 and Y 󸀠 contains elements y1 , . . . , yn+1 such that ‖yi ‖ ∼α c and ‖yi ‖p ∼α e for all 1 ≤ i ≤ n + 1. Since Y contains only n such elements, one of them must be in Y 󸀠 \ Y. To prove the classification theorem, we first need an extension theorem for partial decomposition bases. Lemma 5.4.11 (Jacoby–Leistner–Loth–Strüngmann [76]). Let G and H be groups with rich partial decomposition bases 𝒞 and 𝒟, respectively. Suppose α is an ordinal such that ŵ α (c, p, e, G) = ŵ α (c, p, e, H) for every compatibility class c of Ulm matrices, prime p, and equivalence class e of Ulm sequences. Assume X ∪ {x} ∈ 𝒞 and Y ∈ 𝒟 such that ŵ α (c, p, e, ⟨X⟩0 ) = ŵ α (c, p, e, ⟨Y⟩0 ) for all c, p, and e. Then there exists an element y ∈ H such that Y ∪ {y} ∈ 𝒟 and 0

0

ŵ α (c, p, e, ⟨X ∪ {x}⟩ ) = ŵ α (c, p, e, ⟨Y ∪ {y}⟩ ) for all c, p, and e. In fact, ‖x‖ ∼α ‖y‖ and ‖x‖p ∼α ‖y‖p for all primes p. Proof. The case of x ∈ X is trivial, so assume x ∈ G \ X. Let c0 = [‖x‖], p0 an arbitrary prime, and e0 = [‖x‖p0 ]. Then 0

ŵ α (c0 , p0 , e0 , ⟨X ∪ {x}⟩ ) = ŵ α (c0 , p0 , e0 , ⟨X⟩0 ) + 1 > ŵ α (c0 , p0 , e0 , ⟨Y⟩0 ). By Corollary 5.4.10, there is a Y 󸀠 ∈ 𝒟 such that Y ⊆ Y 󸀠 , and a z ∈ Y 󸀠 \ Y with ‖z‖ ∼α c0 and ‖z‖p0 ∼α e0 . Then ‖x‖ ∼α ‖z‖, and that implies that ‖x‖p =α ‖z‖p for almost all primes p, so we may assume that ‖x‖p ∼α ‖z‖p for all primes p, except {p1 , . . . , pn }.

5.4 Classification up to an ordinal | 229

̃ We will prove, by induction on n, that if there is a z̃ such that Y ∪{z}̃ ∈ 𝒟, ‖x‖ ∼α ‖z‖, and ‖x‖p ∼α ‖z‖̃ p for all but n values of p, then there is a y ∈ H \ Y such that Y ∪ {y} ∈ 𝒟,

‖y‖ ∼α c0 ,

and ‖x‖p ∼α ‖y‖p

for all primes p.

Suppose n = 0. Since 𝒟 is rich, Y ∪ {z} ∈ 𝒟, so we simply take y = z. Now assume the assertion is true for n − 1. Let en = [‖x‖pn ]. By assumption, ‖z‖pn ≁α ‖x‖pn ; that is, ‖z‖pn ≁α en , and so 0

0

ŵ α (c0 , pn , en , ⟨Y ∪ {z}⟩ ) = ŵ α (c0 , pn , en , ⟨Y⟩0 ) < ŵ α (c0 , pn , en , ⟨X ∪ {x}⟩ ). By Corollary 5.4.10, there is an element z 󸀠 ∈ H such that Y ∪ {z, z 󸀠 } ∈ 𝒟, ‖z 󸀠 ‖ ∼α c0 , and ‖z 󸀠 ‖pn ∼α en . Now apply Lemma 5.4.7 to the group ⟨Y ∪ {z, z 󸀠 }⟩0 with decomposition basis Y ∪ {z, z 󸀠 }, the elements z and z 󸀠 , and the prime pn . This then gives us y, y󸀠 ∈ ⟨Y ∪ {z, z 󸀠 }⟩ such that ⟨Y ∪ {z, z 󸀠 }⟩ = ⟨Y ∪ {y, y󸀠 }⟩, ‖y‖pn =α ‖z 󸀠 ‖pn , ‖y󸀠 ‖pn =α ‖z‖pn , and for all p ≠ pn , ‖y‖p =α ‖z‖p , ‖y󸀠 ‖p =α ‖z 󸀠 ‖p . In particular, ‖y‖pn ∼α ‖z 󸀠 ‖pn ∼α en = [‖x‖pn ], and for all p ∉ {p0 , . . . , pn }, ‖y‖p ∼α ‖z‖p ∼α ‖x‖p . Furthermore, ‖y‖p0 =α ‖z‖p0 ∼α ‖x‖p0 , so ‖y‖p ∼α ‖x‖p for all p ∉ {p1 , . . . , pn−1 }. By the induction hypothesis, there is a ỹ such that Y ∪ {y}̃ ∈ 𝒟, ‖y‖̃ ∼α c0 ∼α ‖x‖, and ‖x‖p ∼α ‖y‖̃ p for all primes p. This ỹ is the desired y. We will use the local classification theorem to prove the global one. First, we relate the local and global Warfield invariants. Lemma 5.4.12. Let G and H be PDB groups, p a prime, α an ordinal, and e an equivalence class of Ulm sequences. If ŵ α (c, p, e, G) = ŵ α (c, p, e, H) for all compatibility classes c of Ulm matrices, then ŵ α (e, Gp ) = ŵ α (e, Hp ). Proof. Let C be a collection of representatives from the α-compatibility classes, with exactly one from each class. Using Lemma 5.3.4, we obtain ̂ 󸀠 , Gp )) ∧ ω = ( ∑ ∑ w(c, ̂ p, e󸀠 , G)) ∧ ω ŵ α (e, Gp ) = ( ∑ w(e e󸀠 ∼α e c

e󸀠 ∼α e

̂ 󸀠 , p, e󸀠 , G)) ∧ ω = ( ∑ ∑ ∑ w(c e󸀠 ∼α e c∈C c󸀠 ∼α c

̂ 󸀠 , p, e󸀠 , G)) ∧ ω = ( ∑ ∑ ∑ w(c c∈C e󸀠 ∼α e c󸀠 ∼α c

= ( ∑ ŵ α (c, p, e, G)) ∧ ω. c∈C

By the same argument, we get ŵ α (e, Hp ) = ( ∑ ŵ α (c, p, e, H)) ∧ ω = ( ∑ ŵ α (c, p, e, G)) ∧ ω = ŵ α (e, Gp ). c∈C

c∈C

230 | 5 Groups with partial decomposition bases Now we are ready for the global classification theorem up to λ. Theorem 5.4.13. Let G and H be PDB groups, and let λ be an ordinal. Suppose the following: 1. û p (α, G) = û p (α, H) for all primes p and α < ωλ; 2. ŵ ω(ν+1) (c, p, e, G) = ŵ ω(ν+1) (c, p, e, H) for every compatibility class c of Ulm matrices, prime p, equivalence class e of Ulm sequences, and ν < λ; 3. if length(t(Gp )) < ωλ, then û p (∞, G) = û p (∞, H). Then G ≡λ H. Proof. Let 𝒞 and 𝒟 be rich partial decomposition bases for G and H, respectively. By Theorem 5.1.11, it suffices to prove that the following conditions hold for any ν < λ and any ω(ν + 1)-height-preserving isomorphism f : S → T, where S ⊆ G and T ⊆ H are finitely generated: (a) If X ⊆ S ⊆ ⟨X⟩0 for some X in 𝒞 , where f (X) ∈ 𝒟, and if X ∪ {x} ∈ 𝒞 for some x ∈ G, then there is a nonzero multiple x󸀠 of x and a y󸀠 ∈ H such that f (X) ∪ {y󸀠 } ∈ 𝒟 and g : ⟨S, x󸀠 ⟩ → ⟨T, y󸀠 ⟩, extending f by x󸀠 󳨃→ y󸀠 , is an isomorphism that preserves heights up to ω(ν + 1); (b) If a ∈ G has a nonzero multiple in S, then there is a b ∈ H such that g : ⟨S, a⟩ → ⟨T, b⟩, extending f by a 󳨃→ b, is an isomorphism that preserves heights up to ων; and analogous conditions on partial isomorphisms from H to G. Therefore, suppose f : S → T is an ω(ν +1)-height-preserving isomorphism, where ν < λ, and S ⊆ G and T ⊆ H are finitely generated. Case (a): Suppose for some X ∈ 𝒞 , X ⊆ S ⊆ ⟨X⟩0 , Y = f (X) ∈ 𝒟, and X ∪ {x} ∈ 𝒞 . Since the case of x ∈ S is trivial, we may assume x ∈ ̸ S. Then ŵ ω(ν+1) (c, p, e, ⟨X⟩0 ) = ŵ ω(ν+1) (c, p, e, ⟨Y⟩0 ) for all c, p, and e since f is ω(ν + 1)-height-preserving. Then by Lemma 5.4.11, there is a y ∈ H such that Y ∪ {y} ∈ 𝒟 and ŵ ω(ν+1) (c, p, e, ⟨X ∪ {x}⟩0 ) = ŵ ω(ν+1) (c, p, e, ⟨Y ∪ {y}⟩0 ) for all c, p, e. In fact, ‖x‖ ∼ω(ν+1) ‖y‖ and ‖x‖p ∼ω(ν+1) ‖y‖p for all p. Since ‖x‖ ∼ω(ν+1) ‖y‖, ‖x‖p =ω(ν+1) ‖y‖p for all but finitely many primes p. This means that there are positive integers k and l such that ‖kx‖p =ω(ν+1) ‖ly‖p for all primes p. Let x󸀠 = kx, and y󸀠 = ly. As in the proof of the global classification in L∞ω , ̃ = ⟨S ∩ X⟩, ̃ so by Lemma 5.3.7, there is a positive integer we let X̃ = X ∪ {x󸀠 }. Then S ∩ ⟨X⟩ n such that for all m ∈ ℤ, s ∈ S and p prime, |mnx󸀠 + s|p = |mnx󸀠 |p ∧ |s|p , and similarly for T, |mny󸀠 + t|p = |mny󸀠 |p ∧ |t|p , for all m ∈ ℤ, t ∈ T, and prime p. Now let S󸀠 = ⟨S, nx 󸀠 ⟩, and T 󸀠 = ⟨T, ny󸀠 ⟩. Then extend f to g 󸀠 : S󸀠 → T 󸀠 by sending nx 󸀠 to ny󸀠 . By the choice of n, g 󸀠 is ω(ν + 1)-height-preserving. Since 𝒞 and 𝒟 are rich, X 󸀠 = X ∪ {nx󸀠 } ∈ 𝒞 , and Y 󸀠 = f (X) ∪ {ny󸀠 } ∈ 𝒟. Case (b): Suppose x ∈ G and rx ∈ S for some positive integer r. As in the L∞ω case, we will localize and then use Warfield’s local-global lemma. Let α < ωλ, ν < λ, ̂ Gp ) = u(α, ̂ Hp ) and p a prime, and e an equivalence class of Ulm sequences. Then u(α,

5.4 Classification up to an ordinal | 231

̂ ̂ similarly, u(∞, Gp ) = u(∞, Hp ) if length(t(Gp )) < ωλ (Lemma 1.17.11). By Lemma 5.4.12, we have ŵ ω(ν+1) (e, Gp ) = ŵ ω(ν+1) (e, Hp ). Recall that Gp and Hp have induced partial decomposition bases, say 𝒞Gp and 𝒞Hp , respectively. Hence, Gp and Hp satisfy the conditions of the local Theorem 5.4.6. As part of that proof, we proved an analogous assertion (b), which states, in this case, that if ν < λ, f is an ω(ν + 1)-height-preserving isomorphism, f : S → T, S ⊆ Gp , T ⊆ Hp are finitely generated submodules, and a ∈ Gp has a nonzero multiple in S, then there is a b ∈ H such that g : ⟨S, a⟩ → ⟨T, b⟩, extending f by a 󳨃→ b, is an isomorphism that preserves heights up to ων. Apply this to the induced map fp : Sp → Tp with associated {x ⊗ 1 : x ∈ X} ∈ 𝒞Gp and {y ⊗ 1 : y ∈ Y} ∈ 𝒞Hp . Then fp can be extended to an ων-height-preserving isomorphism g(p) with xp = x ⊗ 1 ∈ domain(g(p)). By the localglobal Lemma 3.4.5, there is a homomorphism g : ⟨S, x⟩ → T 0 , with g(x) = y for some y ∈ T 0 , and gp = g(p) for all p. Since each g(p) is ων-height-preserving and injective for each p, so is g; thus, g : ⟨S, x⟩ → ⟨T, y⟩ is an ων-height-preserving isomorphism. We have not quite finished the classifications in Lλ∞ω . We still need to prove the converse: that λ-equivalence implies that the invariants agree. Recall that Karp’s theorem allows us to take a model-theoretic view of the groups. If G ≡λ H, then any statement of quantifier rank no more than λ that is true in one group will be true in the other. So all we need is such a statement that says, in essence, “ŵ α (c, p, e, G) ≥ n”. The problem is that, as we will see, “decomposition set” is not expressible in L∞ω , so neither is “ŵ α (c, p, e, G) ≥ n”. The solution is to find an equivalent statement that does not refer to decomposition sets. We have already seen that the modified Stanton in̂ p, e, G) agrees with w(c, ̂ p, e, G) whenever G has a decomposition basis. We variant s(c, seek an analogous invariant that agrees with ŵ α (c, p, e, G) and is expressible in L∞ω . Recall the following definitions from Section 4.4. Let G be a group, p a prime, M an Ulm matrix, and α an ordinal: Gα (M) = {x ∈ G : ‖x‖ ≥α M};

Gα (M ∗ )⬦ = {x ∈ Gα (M) : ‖x‖ ≁α M or x is torsion}; 󵄨 󵄨 Gα (Mp∗ , p)⬦ = {x ∈ G : ‖x‖p ≥α Mp and 󵄨󵄨󵄨pi x󵄨󵄨󵄨p =α̸ mp,i for infinitely many i}; Gα (M ∗ , p)⬦ = Gα (M ∗ )⬦ ∪ Gα (Mp∗ , p)⬦ ; Gα (M ∗ , p) = Gα (M) ∩ ⟨Gα (M ∗ , p)⬦ ⟩. Then for c a compatibility class and e an equivalence class, we modified the Stanton invariant defining ̂ (Gα (M)/Gα (M ∗ , p))}, sα̂ (c, p, e, G) = sup{rp,0 where the supremum is taken over all Ulm matrices M with M ∼a c, Mp ∼α e. We proved in Section 4.4 that sα̂ is expressible in L∞ω (Lemma 4.4.7). Now we will prove that it agrees with ŵ α .

232 | 5 Groups with partial decomposition bases Lemma 5.4.14. Let G be a group, α an ordinal, p a prime, and M = [mq,i ] an Ulm matrix. Then we have the following: 1. The map x 󳨃→ px induces an isomorphism f : Gα (M)/Gα (M ∗ , p) → Gα (pM)/Gα ((pM)∗ , p); 2.

If mp,0 ≥ α, then Gα (M) = Gα (pM) and Gα (M ∗ , p) = Gα ((pM)∗ , p).

Proof. The proof of (1) is the same as that for Lemma 3.7.2, with the obvious adjustments for ∼α , =α , and ≥α . To prove (2), suppose mp,0 ≥ α. Then, for all i, mp,i+1 ≥ mp,i ≥ α, so mp,i+1 =α mp,i , and Mp =α (pM)p . Furthermore, Mq = (pM)q for all q ≠ p, so M =α pM. Then ‖x‖ ≥α M if and only if ‖x‖ ≥α pM, and so it follows from the definitions that Gα (M) = Gα (pM), Gα (M ∗ )⬦ = Gα ((pM)∗ )⬦ , and Gα (Mp∗ , p)⬦ = Gα ((pM)∗p , p)⬦ . Applying the lemma repeatedly gives the following: Corollary 5.4.15. Gα (M)/Gα (M ∗ , p) ≅ Gα (pk M)/Gα ((pk M)∗ , p) for every group G, prime p, ordinal α, k ≥ 0, and Ulm matrix M. In Section 3.7, we saw that Warfield and Stanton invariants agree for groups with decomposition bases. Here we wish to prove the analogous result for PDB groups up to a given ordinal α. Note that then the result for PDB groups up to partial isomorphism is a simple corollary. We cannot use that previous result (Theorem 3.7.3) here, since there is no easy relationship between sα̂ and s.̂ In fact, we cannot even use its proof, since it depends on results about ∗-decomposition sets. However, the following lemma, which is analogous to some of the ∗-decomposition results, will be useful. Lemma 5.4.16. Suppose G is a group with partial decomposition basis 𝒞 , α an ordinal, p a prime, M an Ulm matrix, x1 , . . . , xn ∈ X for some X ∈ 𝒞 , where for each i, xi ∈ Gα (M), ‖xi ‖ ∼α M, and ‖xi ‖p =α Mp . If a1 x1 + ⋅ ⋅ ⋅ + an xn ∈ Gα (M ∗ , p) with a1 , . . . , an ∈ ℤ, then ai xi ∈ Gα (M ∗ , p)⬦ for all i = 1, . . . , n. Proof. Suppose a1 x1 + ⋅ ⋅ ⋅ + an xn = z1 + ⋅ ⋅ ⋅ + zm for some z1 , . . . , zm ∈ Gα (M ∗ , p)⬦ . Let z ∈ {z1 , . . . , zm }. Since 𝒞 is a partial decomposition basis for G, we may write pk az = b1 x1 + ⋅ ⋅ ⋅ + bn xn + y, for some integers a, b1 , . . . , bn , with p ∤ a, and y ∈ ⟨Y \ {x1 , . . . , xn }⟩ for some Y ∈ 𝒞 with X ⊆ Y. In fact, we may choose pk , a, and Y such that this holds for all z ∈ {z1 , . . . , zm }. We claim that pk az = pk+1 x + y for some x ∈ ⟨x1 , . . . , xn ⟩ and y ∈ ⟨Y \ {x1 , . . . , xn }⟩. First, suppose z ∈ Gα (Mp∗ , p)⬦ . Then ‖z‖p ≥α Mp and |pi z|p =α̸ mp,i for infinitely many i. In particular, mp,i < α for all i. For any j0 < ω, we may find j such that j + k ≥ j0 and |pj+k z|p =α̸ mp,j+k . Then for all i, |pj+k xi |p = mp,j+k < |pj pk az|p ≤ |pj bi xi |p , since X is a decomposition set. It then follows that pk+1 | bi for each i, proving the claim in this case.

5.4 Classification up to an ordinal | 233

Next, suppose z ∈ Gα (M ∗ )⬦ . If z is torsion, then pk az = 0, proving the claim. Suppose ‖z‖ ≁α M. Suppose further that bi ≠ 0 for some i. Then since ‖xi ‖ ∼α M, for some r > 0, rM ≥α ‖bi xi ‖ ≥ ‖pk az‖ ≥ ‖z‖ ≥α M, contradicting ‖z‖ ≁α M. Hence, bi = 0 for all i, and pk az ∈ ⟨Y \ {x1 , . . . , xn }⟩. Again the claim is satisfied. We then get pk a(a1 x1 + ⋅ ⋅ ⋅ + an xn ) = pk+1 x + y for some x ∈ ⟨x1 , . . . , xn ⟩ and y ∈ ⟨Y \ {x1 , . . . , xn }⟩, since this is true for each z. Equating xi terms, we get p | ai for each i. First, suppose mp,j < α for all j. Then |pj+k ai xi |p = |pj pk aai xi |p > |pj+k xi |p = mp,j+k for all j, and so ai xi ∈ Gα (M) ∩ Gα (Mp∗ , p)⬦ ⊆ Gα (M ∗ , p)⬦ for each i. Now consider the case in which mp,i ≥ α for some i. Let z ∈ {z1 , . . . , zm }. Then |pk z|p ≥α mp,k ≥ α for each k ≥ i, so |pk z|p =α mp,k , and z ∈ ̸ Gα (Mp∗ , p)⬦ . Therefore, we have z ∈ Gα (M ∗ )⬦ . If z is

torsion, pk az = 0. If ‖z‖ ∼α̸ M, then it follows that ‖bi xi ‖ ∼α̸ M for all 1 ≤ i ≤ n with bi ≠ 0; hence, bi = 0 for all i and pk az ∈ ⟨Y \ {x1 , . . . , xn }⟩. In any case, equating xi terms gives pk aai xi = 0, which means ai xi is torsion, and hence, in Gα (M ∗ , p)⬦ .

Theorem 5.4.17 (Jacoby–Loth [70]). Suppose that G is a PDB group, and α is an ordinal. Then rp,0 (Gα (M)/Gα (M ∗ , p)) =ω ŵ α (c, p, e, G) for every compatibility class c of Ulm matrices, prime p, equivalence class e of Ulm sequences, and M ∈ c with Mp ∈ e. Proof. We may assume that G has a rich partial decomposition basis 𝒞 . Suppose M = [mq,i ] ∈ c and Mp ∈ e. Note that by taking p-multiples, which we are allowed to do by Corollary 5.4.15, we may assume that if mp,i ≥ α for some i, then mp,i ≥ α for all i. First, suppose that ŵ α (c, p, e, G) ≥ n for some n < ω, say there are x1 , . . . , xn ∈ X for some X ∈ 𝒞 and for all 1 ≤ i ≤ n, ‖xi ‖ ∼α c and ‖xi ‖p ∼α e. For each 1 ≤ i ≤ n, ‖xi ‖ ∼α M, so for some mi > 0, ‖mi xi ‖ ≥α M. Then if we let m = ∏ni=1 mi , we get ‖mxi ‖ ≥α M for all 1 ≤ i ≤ n. Also, ‖mxi ‖p ∼ ‖xi ‖p ∼α Mp , so there are ki and ki󸀠 such

that ‖pki mxi ‖p =α pki Mp . Let k 󸀠 = ∑ni=1 ki󸀠 and li = ki + ∑j=i̸ kj󸀠 for each i. Then it is easily 󸀠

verified that ‖pli mxi ‖p =α (pk M)p , and for q ≠ p, ‖pli mxi ‖q ≥α (pk M)q . Since 𝒞 is rich, we may use {pli mxi : 1 ≤ i ≤ n} for the decomposition set and write xi in place of pli mxi . 󸀠 Furthermore, by Corollary 5.4.15, it suffices to prove the assertion for pk M, so we may 󸀠 write M in place of pk M. Hence, we may assume xi ∈ Gα (M) and ‖xi ‖p =α Mp for all 1 ≤ i ≤ n. Now we will prove that x1 , . . . , xn are independent representatives of Gα (M) over Gα (M ∗ , p), and of either p-power or infinite order. Suppose a1 x1 +⋅ ⋅ ⋅+an xn ∈ Gα (M ∗ , p). Then ai xi ∈ Gα (M ∗ , p) by Lemma 5.4.16. It remains to prove that the order of each xi + Gα (M ∗ , p) is a p-power or infinite. First, suppose mp,k < α for all k. For each k, since mp,k < α and |pk pxi |p > |pk xi |p = mp,k , we get |pk pxi |p >α mp,k , and thus, pxi ∈ Gα (Mp∗ , p)⬦ ⊆ Gα (M ∗ , p). Suppose xi ∈ Gα (M ∗ , p). Then by the lemma, xi ∈ Gα (M ∗ , p)⬦ , a contradiction since ‖xi ‖p =α Mp , ‖xi ‖ ∼α M, and xi is not torsion. This proves that each xi + Gα (M ∗ , p) has order p; thus, rp (Gα (M)/Gα (M ∗ , p)) ≥ n in that case. Now suppose mp,k ≥ α for all k. We claim the order of each xi + Gα (M ∗ , p) is infinite. Suppose mxi ∈ Gα (M ∗ , p) for some m ≠ 0. Then mxi ∈ Gα (M ∗ , p)⬦ , a contradiction since xi is not 󸀠

󸀠

234 | 5 Groups with partial decomposition bases torsion, ‖mxi ‖ ∼α M, and for all j, |pj mxi | ≥α mp,j ≥ α; hence, |pj mxi |p =α mp,j . Thus, we have r0 (Gα (M)/Gα (M ∗ , p)) ≥ n. This proves that rp,0 (Gα (M)/Gα (M ∗ , p)) ≥ ŵ α (c, p, e, G). Now suppose ŵ α (c, p, e, G) = n < ω, and {x1 , . . . , xn } ∈ 𝒞 is a maximal set with ‖xi ‖ ∼α M and ‖xi ‖p ∼α Mp for all i = 1, . . . , n. As before, we may assume xi ∈ Gα (M) and ‖xi ‖p =α Mp for all i. By the paragraph above, x1 , . . . , xn are independent representatives of Gα (M)/Gα (M ∗ , p). We will show that {x1 + Gα (M ∗ , p), . . . , xn + Gα (M ∗ , p)} is a maximal independent set of elements of p-power or infinite order in Gα (M)/Gα (M ∗ , p). Suppose to the contrary that x1 , . . . , xn , y are independent representatives, where y + Gα (M ∗ , p) has either p-power or infinite order. Choose Y ∈ 𝒞 such that X ⊆ Y and y ∈ ⟨Y⟩0 . Then write ay = a1 x1 + ⋅ ⋅ ⋅ + an xn + b1 y1 + ⋅ ⋅ ⋅ + bm ym , where a > 0 and y1 , . . . , ym ∈ Y \ {x1 , . . . , xn }. We may write a = pk a󸀠 , where p ∤ a󸀠 . Then since Y is a decomposition set and ‖y‖ ≥α M, we get pk M ≤α ‖ay‖ ≤ ‖ai xi ‖ for all i, and similarly for bj yj , so ai xi and bj yj are in Gα (pk M) for all 1 ≤ i ≤ n and 1 ≤ j ≤ m. Now for each 1 ≤ i ≤ n with ai ≠ 0, write ai = pk(ai ) a󸀠i , where p ∤ a󸀠i if a󸀠i ≠ 0. If a󸀠i = 0, let k(ai ) = k. Let

k 󸀠 = min{k, k(a1 ), . . . , k(an )}. By Corollary 5.4.15, pk y, pk x1 , . . . , pk xn are independent 󸀠 󸀠 representatives of Gα (pk M)/Gα ((pk M)∗ , p) ≅ Gα (M)/Gα (M ∗ , p). 󸀠 Now we will prove that b1 y1 + ⋅ ⋅ ⋅ + bm ym ∈ Gα ((pk M)∗ , p). Consider yi for some 1 ≤ i ≤ m. Since {x1 , . . . , xn , bi yi } ∈ 𝒞 , since 𝒞 is rich, and {x1 , . . . , xn } is maximal, we have 󸀠 either ‖bi yi ‖ ≁α c or ‖bi yi ‖p ≁α e. First suppose ‖bi yi ‖ ≁α c. Then ‖bi yi ‖ ≁a M ∼ pk M, 󸀠

󸀠

󸀠

so bi yi ∈ Gα ((pk M)∗ )⬦ . 󸀠 Now suppose ‖bi yi ‖p ≁α e. Then ‖bi yi ‖p ≁α Mp ∼ pk Mp . Suppose there is some 󸀠

j0 such that |pj bi yi |p =α mp,j+k󸀠 for all j ≥ j0 . Then ‖pj0 bi yi ‖p =α pj0 +k Mp , and so, 󸀠

pk Mp ∼α ‖bi yi ‖p ≁α e ∼ Mp ∼ pk Mp , a contradiction. This gives us |pj bi yi |p =α̸ mp,j+k󸀠 󸀠

󸀠

for infinitely many j and bi yi ∈ Gα ((pk M)∗p , p)⬦ . In either case, bi yi ∈ Gα ((pk M)∗ , p). 󸀠

󸀠

Then ay + Gα ((pk M)∗ , p) = a1 x1 + ⋅ ⋅ ⋅ + an xn + Gα ((pk M)∗ , p). By the indepen󸀠 dence assumption, we then have ay, a1 x1 , . . . , an xn ∈ Gα ((pk M)∗ , p). First, suppose 󸀠 󸀠 k 󸀠 = k(ai ) for some i with ai ≠ 0. Then pk a󸀠i xi ∈ Gα ((pk M)∗ , p). By Corollary 5.4.15, a󸀠i xi ∈ Gα (M ∗ , p); thus, a󸀠i xi ∈ Gα (M ∗ , p)⬦ by Lemma 5.4.16 since ‖a󸀠i xi ‖ ∼ ‖xi ‖ ∼α M. But then we get a contradiction since a󸀠i xi is not torsion, ‖a󸀠i xi ‖p = ‖xi ‖p =α Mp , and ‖a󸀠i xi ‖ ∼α M. Thus, rp,0 (Gα (M)/Gα (M ∗ , p)) = n, and the assertion follows. Similarly, if k 󸀠 = k, then y ∈ Gα (M ∗ , p), also a contradiction. 󸀠

󸀠

Corollary 5.4.18. Suppose G is a PDB group. Then sα̂ (c, p, e, G) = ŵ α (c, p, e, G) for all α, c, p, and e. ̂ p, e, G) = w(c, ̂ p, e, G) for all c, p, Corollary 5.4.19. Suppose G is a PDB group. Then s(c, and e. Proof. Choose α = sup{p-length(G) : p prime}. Now we see that for PBD groups, the modified Stanton invariant is independent of the choice of M.

5.5 PDB groups and k-groups | 235

Corollary 5.4.20. Let G be a PDB group and α an ordinal. Then sα̂ (c, p, e, G) = ̂ (Gα (M)/Gα (M ∗ , p)) for all M ∈ c with Mp ∈ e. rp,0 Theorem 5.4.21 (Jacoby–Loth [70]). Suppose G and H are PDB groups. If G ≡λ H, where λ = ωγ for some γ a limit ordinal, then for all α < ωλ, ŵ α (c, p, e, G) = ŵ α (c, p, e, H) for all c, p, e. Proof. Suppose for some c, p, e, and α < ωλ, ŵ α (c, p, e, G) ≥ m. Then by Corollary 5.4.18, sα̂ (c, p, e, G) ≥ m. Write α as ωδ + n and δ as ωδ󸀠 + n󸀠 (n, n󸀠 < ω). By Lemma 4.4.7, there is a formula φ of quantifier rank ≤ δ + ω + m such that G 󳀀󳨐 φ if and only if sα̂ (c, p, e, G) ≥ m. Note that δ󸀠 < γ, so δ󸀠 + 3 < γ, since γ is a limit ordinal. Then λ = ωγ > ω(δ󸀠 + 3) > ωδ󸀠 + n󸀠 + ω + m = δ + ω + m. Since sα̂ (c, p, e, G) ≥ m, G 󳀀󳨐 φ. Since the quantifier rank of φ is < λ and G ≡λ H, we get H 󳀀󳨐 φ and ŵ α (c, p, e, H) ≥ m. By symmetry ŵ α (c, p, e, G) = ŵ α (c, p, e, H). Corollary 5.4.22. Let G and H be PDB groups such that G ≡λ H, where λ = ωγ and γ is a limit ordinal. Then the following hold: 1. û p (α, G) = û p (α, H) for all primes p and α < ωλ; 2. ŵ ω(ν+1) (c, p, e, G) = ŵ ω(ν+1) (c, p, e, H) for every compatibility class c of Ulm matrices, prime p, equivalence class e of Ulm sequences, and ν < λ; 3. If length(t(Gp )) < ωλ, then û p (∞, G) = û p (∞, H). Proof. (1) and (3) follow from Theorem 4.5.7, and (2) follows from Theorem 5.4.21 by noting that ν + 1 < λ since λ is a limit ordinal. Note that this attempt at a converse of Theorem 5.4.13 has been stated and proved only for certain λ. This is to be expected since the converse part of Theorem 4.5.7, on which this is based, applies only to limit ordinals. It is an open question whether this corollary can be broadened. (See Problem 2.) Note that one direction of the global classification theorem 5.3.8 follows easily: ̂ p, e, G) = w(c, ̂ p, e, H) Corollary 5.4.23. Let G and H be PDB groups. If G ≡∞ H, then w(c, for all c, p, e.

5.5 PDB groups and k-groups Now we examine the class of PDB groups that we have just classified. First, we recall the following definitions and facts from Chapter 3: A direct sum ⨁i∈I Ai ⊆ G is said to be a valuated coproduct in G if for all primes p and ordinals α, we have (⨁i∈I Ai ) ∩ pα G = ⨁i∈I (Ai ∩ pα G). We call the valuated coproduct A = ⨁i∈I Ai in G a ∗-valuated coproduct in G if for each Ulm matrix M, Ulm sequence α, prime p, and group F ∈ {G(α∗ , p), G(M ∗ ), G(M ∗ , p)}, A ∩ F = ⨁i∈I (Ai ∩ F). An element x in a group G is called primitive if for any Ulm matrix M, prime p, and positive integer

236 | 5 Groups with partial decomposition bases n, nx ∈ G(M ∗ , p) implies that ‖x‖ ≁ M or nx ∈ G(Mp∗ , p)⬦ . A decomposition set X in G is called a ∗-decomposition set if ⟨X⟩ = ⨁x∈X ⟨x⟩ is a ∗-valuated coproduct in G and each element x ∈ X is primitive. A subgroup A of a group G is called knice if for every finite subset S of G, there is a finite (possibly empty) set of primitive elements x1 , . . . , xn ∈ G such that B = A ⊕ ⟨x1 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨xn ⟩ is a ∗-valuated coproduct in G with ⟨S, B⟩/B finite. If 0 is a knice subgroup of G, then G is called a k-group. By Corollary 3.10.1, if G has a decomposition basis X, then X is a ∗-decomposition set. Recall also that, by Theorem 3.12.3, if B = A ⊕ ⟨x1 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨xn ⟩ is a ∗-valuated coproduct in a group G such that if A is knice in G and each element xi is primitive, then B is knice in G. The relationship between k-groups and PDB groups was first explored in the local case by Jacoby and Loth [71] and then extended to the global case in [73], from which these results were taken. For any group G, we define 𝒞G = {Y : Y is a finite ∗-decomposition set in G}.

Lemma 5.5.1. If G is a k-group, then 𝒞G is a partial decomposition basis for G. Proof. First, 𝒞G ≠ 0 since 0 ∈ 𝒞G . Each ∗-decomposition set is a decomposition set, so we only need to verify the extension property. Suppose X = {x1 , . . . , xn } ∈ 𝒞G and x ∈ G. By Theorem 3.12.3 with A = 0, ⟨x1 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨xn ⟩ is knice in G. By the definition of knice, there is a subset Y = {y1 , . . . , ym } of G such that each yi is primitive and ⟨x1 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨xn ⟩ ⊕ ⟨y1 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨ym ⟩ is a ∗-valuated coproduct with x ∈ ⟨X ∪ Y⟩0 . Thus, X ∪ Y ∈ 𝒞G . Lemma 5.5.2 (Jacoby–Loth [71]). Let G be a group with partial decomposition basis 𝒞 , X ∈ 𝒞 , and {x1 , . . . , xn } ⊆ X. Suppose y = a1 x1 + ⋅ ⋅ ⋅ + an xn , where a1 , . . . , an are nonzero integers, and y = z1 + ⋅ ⋅ ⋅ + zm for some z1 , . . . , zm ∈ G, and let p be a prime. Then for every 1 ≤ j ≤ n, there is 1 ≤ i0 ≤ m such that zi0 is not torsion, and 1. for any Ulm sequence α, if y ∈ G(α, p) and zi0 ∈ G(α∗ , p)⬦ , then aj xj ∈ G(α∗ , p)⬦ ; 2. for every Ulm matrix M, if y ∈ G(M) and zi0 ∈ G(M ∗ )⬦ , then aj xj ∈ G(M ∗ )⬦ . Proof. Let Y ∈ 𝒞 such that X ⊆ Y and z1 , . . . , zm ∈ ⟨Y⟩0 . We apply the analogous lemma for decomposition bases, Lemma 3.6.3 to H = ⟨Y⟩0 , giving an 1 ≤ i0 ≤ m for each 1 ≤ j ≤ n as in that lemma. Then if α is an Ulm sequence, y ∈ G(α, p), and zi0 ∈ G(α∗ , p)⬦ , then since H is isotype in G, we have y ∈ H(α, p), zi0 ∈ H(α∗ , p)⬦ ; and thus, aj xj ∈ H(α∗ , p)⬦ ⊆ G(α∗ , p)⬦ . The second part is similar. Lemma 5.5.3 (Jacoby–Loth [71]). Suppose G is a group with partial decomposition basis 𝒞 . Let X ∈ 𝒞 , x1 , . . . , xn ∈ X, a1 , . . . , an nonzero integers, M an Ulm matrix, α an Ulm sequence, p a prime, and F ∈ {G(α∗ , p), G(M ∗ ), G(M ∗ , p)}. If a1 x1 + ⋅ ⋅ ⋅ + an xn ∈ F, then ai xi ∈ F ⬦ for all 1 ≤ i ≤ n. Proof. The result follows from Lemma 5.5.2 exactly as Lemma 3.6.4 followed from Lemma 3.6.3.

5.5 PDB groups and k-groups | 237

Lemma 5.5.4. Suppose G is a group with partial decomposition basis 𝒞 . If X ∈ 𝒞 , then X is a ∗-decomposition set in G. Proof. Let X ∈ 𝒞 , x ∈ X, p prime, α an Ulm sequence, and M an Ulm matrix. Since X is a decomposition set, the elements are independent. To prove they are primitive, suppose kx ∈ G(M ∗ , p) for some x ∈ X and positive integer k. By Lemma 5.5.3, kx ∈ G(M ∗ , p)⬦ . If kx ∈ G(M ∗ )⬦ , then ‖x‖ ∼ ‖kx‖ ≁ M. If not, kx ∈ (Mp∗ , p)⬦ . Now we wish to show that ⟨X⟩ = ⨁x∈X ⟨x⟩ is a ∗-valuated coproduct. It is a valuated coproduct since X is a decomposition set. Suppose that F is any group in {G(α∗ , p), G(M ∗ ), G(M ∗ , p)}, and y = a1 x1 + ⋅ ⋅ ⋅ + an xn ∈ F for some x1 , . . . , xn ∈ X and a1 , . . . , an nonzero integers. For any 1 ≤ i ≤ n, ai xi ∈ F ⬦ by Lemma 5.5.3. This proves (⨁x∈X ⟨x⟩) ∩ F ⊆ ⨁x∈X (⟨x⟩ ∩ F), as desired. Theorem 5.5.5 (Jacoby–Loth [71]). A group has a partial decomposition basis if and only if it is a k-group. Proof. Suppose G has a partial decomposition basis 𝒞 , and let S be a finite subset of G. Then there is an X ∈ 𝒞 such that S ⊆ ⟨X⟩0 , say X = {x1 , . . . , xn }. By Lemma 5.5.4, X is a ∗-decomposition set, specifically, x1 , . . . , xn are primitive and ⟨x1 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨xn ⟩ is a ∗-valuated coproduct. This means that 0 is knice, and so G is a k-group. Conversely, if G is a k-group, then it has a partial decomposition basis 𝒞G by Lemma 5.5.1. Now we may use what we know about k-groups to answer a long-standing question about PDB groups. Corollary 5.5.6. The class of PDB groups is closed under direct summands. Proof. By Theorem 3.12.11, a direct summand of a k-group is a k-group. Likewise, we may use what we know about PDB groups to shed light on k-groups. Corollary 5.5.7. Every finitely generated subgroup of a k-group is locally nice. Proof. By Corollary 5.1.8, this is true for PDB groups. We have often expanded a given partial decomposition basis to give us what we need to complete a proof or construction. Now we see that there is a largest partial decomposition basis, and, furthermore, we know what it looks like. Proposition 5.5.8. Suppose that G is a group with partial decomposition basis 𝒞 . Then the set 𝒞G of all finite ∗-decomposition sets in G is a partial decomposition basis for G, and we have 𝒞 ⊆ 𝒞G . Proof. Suppose G has a partial decomposition basis 𝒞 . Then G is a k-group by Theorem 5.5.5, and it has a partial decomposition basis 𝒞G = {Y : Y is a finite ∗-decomposition set in G} by Lemma 5.5.1. Let X ∈ 𝒞 . Then by Lemma 5.5.4, X is a ∗-decomposition set in G, and so X ∈ 𝒞G .

238 | 5 Groups with partial decomposition bases Corollary 5.5.9. Let G be a k-group and X a finite ∗-decomposition set in G. If Y is a finite decomposition set in G such that ⟨Y⟩ = ⟨X⟩, then Y is a ∗-decomposition set in G. Proof. The k-group G has a partial decomposition basis 𝒞G (Lemma 5.5.1). By Lemma 5.1.3, G has a rich partial decomposition basis 𝒞 ⊇ 𝒞G . But by Proposition 5.5.8, 𝒞 ⊆ 𝒞G ; hence, 𝒞G = 𝒞 is rich. In particular, if X ∈ 𝒞G and ⟨X⟩ = ⟨Y⟩, then Y ∈ 𝒞G .

5.6 Partial subbases and properties of ∗-decomposition sets We often take advantage of isotype subgroups to transfer characteristics of the larger group to the subgroup. For example, if 𝒞 is a partial decomposition basis and X ∈ 𝒞 , then X is a decomposition basis for ⟨X⟩0 , which allows us to apply what we know about decomposition bases to partial decomposition bases. We would have liked to do something similar to prove Lemma 5.5.4, using the analogous Lemma 3.10.1 for decomposition bases, but we cannot argue that X a ∗-decomposition set in ⟨X⟩0 implies that it is one in all of G. The following example shows a set that is a ∗-decomposition set in H, an isotype subgroup of G, but not a ∗-decomposition set in G. We will be exploring which properties do and do not transfer between a group and an isotype subgroup. Example 5.6.1 (Hill–Megibben [49]). There is a group G, an isotype subgroup H of G, and an X ⊆ H such that X is a finite ∗-decomposition set in H, but not in G. Proof. Define Ha , Hb , and Hc as in Corollary 2.7.5, with ‖a‖p = (0, 3, 4, 7, 8, 11, 12, . . . ) and ‖b‖p = (1, 2, 5, 6, 9, 10, . . . ) for some prime p, and ‖c‖q = (|qi a|q ∧ |qi b|q )i 0, rzi ∈ B = H ⊕ A for all 1 ≤ i ≤ n. Let 1 ≤ i ≤ n. Then rzi = yi + ai , for some yi ∈ H and ai ∈ A. Now we define rFG⬦ , as follows: rFG⬦ = G((rα)∗ , p)⬦ if FG = G(α∗ , p); rFG⬦ = G((rM)∗ )⬦ if FG = G(M ∗ ); and rFG⬦ = G((rM)∗ , p)⬦ if FG = G(M ∗ , p). Since zi ∈ FG⬦ , it may be verified that rzi ∈ rFG⬦ . Furthermore, since ‖yi ‖ ≥ ‖rzi ‖, yi ∈ rFG⬦ . In fact, yi ∈ rFG⬦ ∩ H = rFH⬦ , since H is isotype. Returning to h, we have rh = rz1 + ⋅ ⋅ ⋅ + rzn = y1 + ⋅ ⋅ ⋅ + yn + a1 + ⋅ ⋅ ⋅ + an = y1 + ⋅ ⋅ ⋅ + yn ∈ rFH . If FH = H(M ∗ , p), then Section 3.6(c), (d) say that rh ∈ H((rM)∗ , p) = rH(M ∗ , p) implies h ∈ H(M ∗ , p). Similarly, we may prove the analogous statements for the other possibilities of F. Thus, h ∈ FH , proving FG ∩ H ⊆ FH . The other inclusion is obvious. Theorem 5.6.17. Let G be a group, H a knice isotype subgroup of G, p a prime, α an Ulm sequence, M an Ulm matrix, and FG ∈ {G(α∗ , p), G(M ∗ ), G(M ∗ , p)}. Then the following hold: 1. FH = FG ∩ H. 2. If x is primitive in H, it is primitive in G. 3. If X is a ∗-decomposition set in H, it is one in G. 4. If A = ⨁i∈I Ai is a ∗-valuated coproduct in H, it is one in G. Proof. Since H is knice and isotype, it is ∗-isotype by Theorem 5.6.16, which gives (1). To prove (2), assume x is primitive in H and nx ∈ G(M ∗ , p) for some n > 0. Then nx ∈ G(M ∗ , p)∩H = H(M ∗ , p), and so, since x is primitive in H, ‖x‖ ≁ M or |pi nx|p ≠ mp,i for infinitely many i < ω. Next, to prove (4), suppose A is a ∗-valuated coproduct in H. Then, since A ⊆ H, A ∩ FG = A ∩ FG ∩ H = A ∩ FH = ⨁i∈I (Ai ∩ FH ) = ⨁i∈I (Ai ∩ FG ). Finally, (3) follows from (2) and (4). Theorem 5.6.18. Let H be an isotype knice subgroup of a group G. Then G is a k-group if and only if H is a k-group. Proof. If G is a k-group, then so is H by Corollary 5.6.11. Now suppose H is a k-group. Let S ⊆ G be finite. Since H is knice in G, there are primitive y1 , . . . , ym ∈ G such that B = H ⊕ ⟨y1 ⟩ ⊕ ⋅ ⋅ ⋅ ⊕ ⟨ym ⟩ is a ∗-valuated coproduct in G and S ⊆ B0 , so r⟨S⟩ ⊆ B for some r > 0. Let S󸀠 = {π(rs) : s ∈ S}, where π : B → H is the projection map, which is a finite subset of H. Since H is a k-group, there is a ∗-decomposition set {x1 , . . . , xn } in H, and hence in G, such that S󸀠 ⊆ ⟨x1 , . . . , xn ⟩0 . It follows that {x1 , . . . , xn , y1 , . . . , ym } is a ∗-decomposition set in G such that S ⊆ ⟨x1 , . . . , xn , y1 , . . . , ym ⟩0 . At this point, we have not resolved the following questions for G a group and H an isotype subgroup: Problem 1. If X ⊆ H is a ∗-decomposition set in G, is it necessarily one in H? Problem 2. If A = ⨁i∈I Ai ⊆ H is a ∗-valuated coproduct in G, is it necessarily one in H?

244 | 5 Groups with partial decomposition bases

5.7 PDB groups and Warfield groups Recall that Warfield groups may be characterized as those groups G such that for some X ⊆ G, 1. X is a decomposition basis for G; 2. ⟨X⟩ is nice in G; 3. G/⟨X⟩ is simply presented. The class of PDB groups is strictly larger than the class of Warfield groups. In fact, for each of these three conditions, we will produce an example of a group that fails the condition and yet is a PDB group. We will give examples that are neither torsion nor torsion-free, and in fact do not split into torsion and torsion-free summands; recall that such mixed groups are said to be nonsplitting. In the following, for a direct summand A of a group G, we let πA : G → A denote the projection map. Theorem 5.7.1 (Jacoby [68]). Let H be a reduced group of torsion-free rank 1, say H/⟨x⟩ is torsion. Let Hi = H for all i < ω. Define a subgroup G of ∏i