COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

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Pergamon Press Ltd., Headington Hill Hall, Oxford 4 & 5 Fitzroy Square, London W.l Pergamon Press (Scotland) Ltd., 2 & 3 Teviot Place, Edinburgh 1 Pergamon Press Inc., Maxwell House, Fairview Park, Elmsford, New York 10523 Pergamon of Canada Ltd., 207 Queen’s Quay West, Toronto 1 Pergamon Press (Aust.) Pty. Ltd., 19a Boundary Street, Rushcutters Bay, N.S.W. 2011, Australia Pergamon Press-S.A.R.L., 24 rue des Ecoles, Paris 5e Vieweg & Sohn GmbH, Burgplatz 1, Braunschweig Copyright @ 1970 Pergamon Press Ltd. All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any term 01 by any means, electronic, mechanical, photocopyIn& recording or otherwise, without the prior permission of Pergamon Press Ltd. First edition 1970 E fy; i; ,il; ..i’ :: :. ._ ,. - . ‘, Library of Congress Catalog Card No. 75-84072 Printed in Hungary 08 012975 7 Foreword Preface Contents J. NEUB~~SER : Investigations of groups on computers 1 J. LEECH: Coset enumeration 21 C. M. CAMPBELL: Some examples using coset enumeration 37 N. S. MENDELSOHN: Defining relations for subgroups of finite index of groups with a finite presentation 43 M. J. DUNWCK~DY: Nielsen transformations 45 H. J~~RGENSEN: Calculation with the elements of a finite group given by generators and defining relations V. FEISCH and J. NEUB~~SER: On a programme for the determination of the automorphism group of a finite group L. GERHARDS and E. ALTMANN: A computational method for determining the automorphism group of a finite solvable group W. L<strong>IN</strong>DENBERG and L. GERHARLM : Combinatorial construction by computer of the set of all subgroups of a finite group by composition of partial sets of its subgroups K. FERBER and H. J~~RGENSEN: A programme for the drawing of lattices 83 J. MCKAY: The construction of the character table of a finite group from generators and relations 89 C. BROTT and J. NEIJB~~SER: A programme for the calculation of characters and representations of finite groups J. S. FRAME: The characters of the Weyl group ES R. Bij~ow and J. NEUB~~SER : On some applications of group-theoretical programmes to the derivation of the crystal classes of & 131 M. HALL JR. : A search for simple groups of order less than one million 137 C. C. SIMS : Computational methods in the study of permutation groups 169 v vii 47 59 61 75 101 111
vi Contents E. KRAUSE and K. WESTON: An algorithm related to the restricted Burnside group of prime exponent 185 A. L. TRITTER: A module-theoretic computation related to the Burnside problem J. J. CANNON: Some combinatorial and symbol manipulation pro- 189 Foreword grams in group theory 199 IF the electronic digital computer has not already virtually taken over the P. G. RUUD and R. -OWN: The computation of irreducible representations of finite groups of order 2”, n e 6 205 world’s arithmetic it will have done so before long. This remarkable success story in the field of applied mathematics-much of it very simple N. S. MENDELSOHN: Some examples of man-machine interaction in the solution of mathematical problems R. J. PLEMMONS: Construction and analysis of non-equivalent finite semigroups 217 223 mathematics, admittedly, such as cost calculations-contrasts strongly with the machine’s very moderate impact so far on pure mathematics. Obvious economic considerations account for much of this difference: no one can be surprised that the need to optimize the design of a nuclear power reactor or to develop a stores holding and purchasing strategy for a T. TAMURA: Some contributions of computation to semigroups and groupoids D. E. KNUTH and P. B. BENDIX: Simple word problems in universal algebras 229 263 large factory provides more economic drive than, say, the properties of the partition function. But this can hardly apply in a university environment, where the pure mathematicians have as much right to the central computing service as anyone else. The explanation must be the great difficulty of the problems; the machine offers great powers of logical processing, of L. J. PAIGB: The application of computers to research in non-associative algebras 299 which arithmetic is only a small and not very interesting part, but it is far from clear how these can be used in the service of anything that can be called genuine mathematics. C. M. GLENN~E: Identities in Jordan algebras 307 A. D. KEEDWELL: On property D neofields and some problems concerning orthogonal latin squares 3 1 5 However, work has in fact been going on in various fields of pure mathematics ever since computers became available. Professor Douglas Munn, surveying the scene from the point of view of an algebraist, decided in 1966 that enough had been done on the application of computers to J. W. P. HIRSCHFELD: A projective configuration W. D. MAURER: The uses of computers in Galois theory 321 325 abstract algebra to warrant the holding of an international conference, to assess what had been achieved and to identify promising lines of future research. A frequent visitor to the Atlas Laboratory, he talked about this with me and my colleague Dr. Robert Churchhouse, and after very little discussion we found ourselves agreeing that the time was indeed right for such a conference, that Oxford would be an admirable location and that the Atlas Laboratory should organize it. We approached Professor Graham Higman who immediately and enthusiastically agreed to give his support; and with equal immediacy and enthusiasm had Professor Coulson’s permission to hold the conference in Oxford’s beautiful new Mathematics Institute. The success of the meeting must be judged from the quality of the papers, reproduced in this volume. I want to record my gratitude to the Science Research Council, who allowed me to meet some of the expenses from Atlas Laboratory funds, and to I.C.T. Limited for generous financial support: jointly, they made it possible to hold the conference. The staff of the Mathematics Institute were as helpful and welcoming as anyone vii J. H. CONWAY: An enumeration of knots and links, and some of their algebraic properties 329 H. F. TROTTER: Computations in knot theory 359 S. L<strong>IN</strong>: Computer experiments on sequences which form integral bases 365 H. COHN: Application of computer to algebraic topology on some bicomplex manifolds 371 H. ZASSENHAUS: A real root calculus 383 R. E. QLMAN : Some computational problems and methods related to invariant factors and control theory 393 List of participants 399
Page 1: COMPUTATIONAL PROBLEMS IN ABSTRACT
Page 5 and 6: X Preface I am indebted to the auth
Page 7 and 8: 4 J. Neubiiser 2.3.5. Added in proo
Page 9 and 10: 8 J. Neubiiser Investigations of gr
Page 11 and 12: 12 J. Neubiiser For 1 es 4 r, ws is
Page 13 and 14: 16 J. Neubiiser Investigations of g
Page 23 and 24: Some examples using coset enumerati
Page 25 and 26: 40 C. M. Campbell Some examples usi
Page 27 and 28: 44 N. S. Mendelsohn for example, in
Page 29 and 30: 48 H. Jiirgensen Calculation with e
Page 35 and 36: 60 V. Fe&h and J. Neubiiser 2. The
Page 37 and 38: 64 L. Gerhards and E. Altmann Autom
Page 39 and 40: 68 L. Gerhards and E. Altmann Autom
Page 41 and 42: 72 L. Gerhards and E. Altmann ni E
Page 43 and 44: 76 W. Lindenberg and L. Gerhards Se
Page 45 and 46: 80 W. Lindenberg and L. Gerhards Se
Page 47 and 48: 84 K. Ferber and H. Jiirgensen The
Page 49 and 50: 7 The construction of the character
Page 51 and 52: 92 John McKay defining multiplicati
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96 John McKay Construction of chara
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100 John McKay In the table, c indi
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104 C. Brott and J. Neubiiser irred
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108 C. Brott and J. Neubiiser eleme
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112 J. S. Frame The characters of t
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116 J. S. Frame 3. The decompositio
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TABLE 5. The Z, character block for
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132 R. Biilow and J. Neubiiser Deri
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A search for simple groups of order
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140 Marshall Hall Jr. Simple groups
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148 Marshall Hall Jr. otherwise no
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152 Marshall Hall Jr. easily found
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160 Marshall Hail Jr. This leads to
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164 Marshall Hall Jr. b= (OO)(Ol, 0
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168 Marshall Hall Jr. 11. R. BRAUER
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172 Charles C. Sims THEOREM 2.3. Th
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176 Charles C. Sims has a set of im
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180 Degrc - No. Order t N (-63 Gene
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An algorithm related to the restric
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A module-theoretic computation rela
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192 A. L. Tritter expressed in the
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196 A. L. Tritter a question is mos
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200 John J. Cannon stack. The Cayle
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, I The computation of irreducible
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208 P. G. Ruud and R. Keown product
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212 P. G. Ruud and R. Keown TX wher
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216 P. G. Ruud and R. Keown 4. L. G
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220 N. S. Mendelsohn where it is kn
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224 Robert J. Plemmons such class [
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228 Robert J. Plemmons ! TOTALS Sem
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232 Takayuki Tamura 8” = 0B if an
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236 Takayuki Tamura Case ,Q = {e).
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240 Takayuki Tamura The calculation
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244 Takayuki Tamura We calculate46
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248 Takayuki Tamura Immediately we
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252 Takayuki Tamura Let U be a subs
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256 Takayuki Tamura TABLE 8. AI1 Se
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I The author and R. Dickinson have
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264 Donald E. Knuth and Peter B. Be
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300 Lowell J. Paige Non-associative
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304 Lowell J. Paige We may lineariz
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308 T(n) U(n) 0) [VI R-4 C. M. Glen
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312 C. M. Glennie where in each cas
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316 A. D. Keedwell of the elements
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A projective coqfiguration J. W. P.
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The uses of computers in Galois the
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328 W. D. Maurer mod p; for a facto
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332 J. H. Conway Enumeration of kno
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J. H. Conway Enumeration of knots a
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340 J. H. Conway -thus our symmetry
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344 J. H. Conway 2 string links to
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348 J. H. Conway 3 and 4 string lin
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352 Non-alternating J. H. Conway L
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356 J. H. Conway Alternating 11 cro
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H. F. Trotter Computations in knot
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364 H. F. Trotter to a. Schubert [1
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368 Shen Lin sequence of squares, t
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372 Harvey Cohn We also consider GZ
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376 Harvey Cohn In the case of Fig.
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380 Harvey Cohn always dictated by
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384 Hans Zassenhaus A real root cal
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388 Hans Zassenhaus the elements A,
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392 Hans Zassenhaus It is clear fro
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396 R. E. Kalman Invariant factors
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400 List of participants DR. J. J.
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