Groups and Computation III: Proceedings of the International Conference at The Ohio State University, June 15-19, 1999 [Reprint 2013 ed.] 9783110872743, 9783110167214

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Ohio State University Mathematical Research Institute Publications 8 Editors: Gregory R. Baker, Walter D. Neumann, Karl Rubin

Ohio State University Mathematical Research Institute Publications 1 2 3 4 5 6 7

Topology '90, B. Apanasov, W. D. Neumann, A. W. Reid, L. Siebenmann (Eds.) The Arithmetic of Function Fields, D. Goss, D. R. Hayes, M. I. Rosen (Eds.) Geometric Group Theory, R. Charney, M. Davis, M. Shapiro (Eds.) Groups, Difference Sets, and the Monster, K. T. Arasu, J. F. Dillon, K. Harada, S. Sehgal, R. Solomon (Eds.) Convergence in Ergodic Theory and Probability, V. Bergelson, R March, J. Rosenblatt (Eds.) Representation Theory of Finite Groups, R. Solomon (Ed.) The Monster and Lie Algebras, J. Ferrar, Κ. Harada (Eds.)

Groups and Computation III Proceedings of the International Conference at The Ohio State University June 15-19, 1999

Editors

William M. Kantor Akos Seress r

W _G DE

Walter de Gruyter · Berlin · New York 2001

Editors William M. Kantor Department of Mathematics, University of Oregon, Eugene, OR 97403-1222, USA Akos Seress Department of Mathematics, The Ohio State University, Columbus, OH 43210-1174, USA Series Editors Gregory R. Baker Department of Mathematics, The Ohio State University, Columbus, OH 43210-1174, USA Karl Rubin Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA Walter D. Neumann Department of Mathematics, Columbia University, New York, NY 10027, USA Mathematics Subject Classification 2000: 20-06; 20-04, 20-B40, 20-D06, 20-D08, 20-F05, 20-G40, 20-P05, 68-W20 © Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.

Library of Congress — Cataloging-in-Publication Data Groups and computation III : proceedings of the international conference at the Ohio State JJniversity, June 15—19, 1999 / editors, William M. Kantor, Akos Seress. p. cm. - (Ohio State University Mathematical Research Institute publications, ISSN 0942-0363 ; 8) ISBN 3-11-016721-2 (alk. paper) 1. Group theory - Data processing - Congresses. 2. Algebra — Data processing — Congresses. I. Title: Groups and computation 3, II. Kantor, W. M. (William M.), 1944- . III. Seress, Akos, 1958- . IV. Series QA174.7.D36 G765 2001 512'.2—dc21 00-065758

Die Deutsche Bibliothek — Cataloging-in-Publication Data Groups and computation III : proceedings of the international conference at the Ohio St^te University, June 1 5 - 19, 1999 / ed. William M. Kantor ; Akos Seress. — Berlin ; New York : de Gruyter, 2001 (Ohio State University Mathematical Research Institute publications ; 8) ISBN 3-11-016721-2

© Copyright 2001 by Walter de Gruyter GmbH & Co., D-10785 Berlin. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Typeset using the authors' T E X files: I. Zimmermann, Freiburg. Printing: WB-Druck GmbH & Co., Rieden/Allgäu. Binding: Lüderitz & Bauer-GmbH, Berlin. Cover design: Thomas Bonnie, Hamburg.

Preface

The conference "Groups and Computation" took place at The Ohio State University in Columbus, Ohio, on June 15-19, 1999. This conference was the successor of two workshops on "Groups and Computation" held at DIM ACS in 1991 and 1995; this is reflected by the "ΠΙ" in the title of the present volume. The main objective of the conference was to provide a forum for merging theory and practice of computations with groups: a continuation of the ongoing dialogue between researchers studying the complexity of group computations and those engaged in the development of practical algorithms. Based on comments of the participants, both during and after the conference, we can safely state that this goal was achieved. There were talks by experts as well as by younger researchers. These proceedings contain accounts of most of the presentations of the various speakers. Nearly half of the participants were from abroad, reflecting the fact that most of the principal researchers in this field were able to attend the conference. In addition to the lectures, the scientific program contained an evening devoted to software demonstrations, another evening for a Magma workshop, and an afternoon for discussion of the future of the matrix group project. Computation with matrix groups has attracted a great deal of significant research in the last few years, and many of the talks were related to various aspects of that subject. Numerous other areas of Computational Group Theory were well-represented: permutation group algorithms, finitely presented groups, polycyclic groups, and applications in science and engineering. We are indebted to Marilyn Radcliff for her invaluable help with the complicated conference logistics. The meeting was generously supported by the Mathematical Research Institute of The Ohio State University, the National Science Foundation, and the National Security Agency. William M. Kantor Ákos Seress

Table of contents

Preface

ν

Christine Altseimer and Alexandre V. Borovik Probabilistic recognition of orthogonal and symplectic groups

1

Roberto M. Avanzi, Mathias Kratzer and Gerhard O. Michler Janko's simple groups J2 and J3 are irreducible subgroups of SLg5(5) with equal centralizers of an involution

21

László Babai and Aner Shalev Recognizing simplicity of black-box groups and the frequency of /7-singular elements in affine groups

39

Robert Beats Improved algorithms for the Tits alternative

63

Peter A. Brooksbank A constructive recognition algorithm for the matrix group Ω (J, q)

79

Peter A. Brooksbank and William M. Kantor On constructive recognition of a black box PSL(d, q)

95

Marston Conder and Charles R. Leedham-Green Fast recognition of classical groups over large

fields

113

Gene Cooperman Parallel GAP: Mature interactive parallel computing

123

Bettina Eick Computing with infinite polycyclic groups

139

Bettina Eick and Alexander Hulpke Computing the maximal subgroups of a permutation group 1

155

Robert M. Guralnick and Frank Lübeck On /^-singular elements in Chevalley groups in characteristic ρ

169

George Havas and Colin Ramsay Experiments in coset enumeration

183

viii

Table of contents

George Havas, Leonard H. Soicher and Robert A. Wilson A presentation for the Thompson sporadic simple group

193

Derek F. Holt Computing automorphism groups of finite groups

201

Christoph Köhler and Herbert Pahlings Regular orbits and the ifc(G V)-problem

209

Charles R. Leedham-Green The computational matrix group project

229

Frank Lübeck Finding //-elements in finite groups of Lie type

249

Alexei D. Miasnikov and Alexei G. Myasnikov Balanced presentations of the trivial group on two generators and the Andrews-Curtis conjecture

257

Takunari Miyazaki Deterministic algorithms for management of matrix groups

265

Peter M. Neumann Nearly simple matrix groups: desiderata and suggestions

281

Peter M. Neumann and Cheryl E. Praeger Cyclic matrices and the MEATAXE

291

Igor Pak What do we know about the product replacement algorithm?

301

Charles C. Sims On the complexity of the endomorphism problem for free groups

349

Talks

361

List of participants

365

Probabilistic recognition of orthogonal and symplectic groups Christine Altseimer* and Alexandre V. Borovik

Abstract. We propose a fast one-sided Monte Carlo algorithm to distinguish, to any given degree of certainty, the symplectic group Cn{q) = PSp2„ iq) from the orthogonal group Bn(q) = 0,2n+i(q) where q > 3 is odd and η and q are given. The algorithm does not use an order oracle and works in polynomial, of η logg, time. 1991 Mathematics Subject Classification: primary 51F15; secondary 20G40.

1. Introduction The aim of this paper is to propose a fast Monte Carlo algorithm to distinguish, to any given degree of certainty, the symplectic group C„(q) = PSp2„ iq) from the orthogonal group Βn(q) = Çl2n+\(q) where q > 3 is odd and η and q are given. Namely we prove the following theorem. Theorem 1.1. There is a polynomial time one-sided Monte Carlo algorithm which, when given a black box group G isomorphic to a simple classical group of type Bn or Cn defined over some field of given size q > 3 and odd characteristic, can decide whether G is isomorphic to Cn or not. If G is Cn then, with probability > 1/2, the algorithm produces a witness against isomorphism G ~ Bn(q). This results answers a question which L. Babai asked the second author. The method of the present paper can be easily extended to the case q = 3 at the expense of adding a few technical details. We decided not to do this because the cases of small fields are covered in the fundamental work by Kantor and Seress [14]. A earlier version of our algorithm has appeared in [2]. It was one-sided only in the case of odd n. Black box groups were introduced by Babai and Szemerédi [5] as the ideal setting for some group computational problems and their importance has grown ever since. For the following compare Babai [4] and Beals [6], * Research supported by the German Academic Exchange Service (DAAD, Doktorandenstipendium HSPII/AUFE). The work on the paper had started when the first author was visiting UMIST.

Groups and Computation ΙΠ Ohio State Univ. Math. Res. Inst. Pubi. 8

©Walter de Gruyter 2001

2

Christine Altseimer and Alexandre V. Borovik

A black-box group G is a group whose elements are encoded as 0-1 strings of uniform length Ν, and the group operations are performed by an oracle (the "black box")· Note that we have the upper bound 2N for |G|. Given strings representing g,h e G, the black box can compute strings representing gh and and decide whether g = h. We denote by μ an upper bound for the time requirement of these group operations, and by ξ an upper bound for the time requirement per element of constructing independent, (nearly) uniformly distributed elements of G. In the following we assume for the sake of argument that the black box gives us independent uniformly distributed elements of G. Compare Kantor and Seress [14] for similar assumptions and Babai [4] for a critical discussion. A Monte Carlo algorithm is a randomised algorithm, where the result is correct with a probability at least 1 — e with 0 < e < 1/2. After repeating a Monte Carlo algorithm several times we can obtain the answer with an arbitrarily small probability of error e. A Monte Carlo algorithm with two possible answers 'yes' and 'no' is one-sided if the answer 'yes' is always correct.

1.1. Why is the case of B„(q) versus Cn(q) so special? Our algorithm deals with one of the more difficult cases in the recognition of black box groups with order oracle. The case of B„(q) versus Cn(q) cannot be handled by a direct study of statistics of orders of elements since these statistics are for all practical purposes the same in both groups. Instead, our algorithm is based on a more detailed study of involutions in these groups. The justification for our approach can be formulated in short as follows: • The simple group of Lie type Bn(q),q odd, cannot be distinguished from Cn (q) by statistics of orders of elements in a random sample of length bounded by a polynomial in nlogq. Indeed, these statistics are controlled by the Dynkin diagram which is the same in these groups. A more detailed explanation is given in Section 1.4. • However, the distribution of involutions in conjugacy classes and the structure of centralisers of involutions are controlled by the extended Dynkin diagrams (Iwahori [13]) which are different for Bn and C„ (Figure 1). Our Monte Carlo algorithm detects the difference in the behaviour of involutions.

c

Figure 1: Extended Dynkin diagram for Bn {left) and Cn (right), η ^ 3.

Probabilistic recognition of orthogonal and symplectic groups 1.2. Description of the algorithm: the easy case, η is odd. Recall that denotes the kernel of the spinor norm S02n+l(#)

3 (q)

> {±1 };

it is a subgroup of index 2 in S02„+i (q) and coincides with the commutator of S02n+i (q) if η ^ 2 and q ^ 3 (for details see, for example, Artin [3]). It is well known that PSp2(