The Primitive Soluble Permutation Groups of Degree less than 256
This monograph addresses the problem of describing all primitive soluble permutation groups of a given degree, with particular reference to those degrees less than 256. The theory is presented in detail and in a new way using modern terminology. A descrip
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Subseries: Australian National Universtiy, Canberra Advisers: L. G. Kovacs, B. H. Neumann and M. F. Newman
1519
M. W. Short
The Primitive Soluble Permutation Groups of Degree less than 256
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Author Mark W. Short Mathematics Programme Murdoch University Murdoch, WA 6150, Australia
Mathematics Subject Classification (1991): 20-04, 20B35, 20D 10
ISBN 3-540-55501-3 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-55501-3 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer- Verlag Berlin Heidelberg 1992 Printed in Germany Typesetting: Camera ready by author/editor Printing and binding: Druckhaus Beltz, HemsbachlBergstr. 46/3140-543210 - Printed on acid-free paper
Contents 1 Introduction 1.1
1
Motivation-a maximal subgroups algorithm
1
1.2
Primitive permutation groups
4
1.3
Summary of contents . . . . .
6
1.4
Conventions and Notation ..
1.5
Polycyclic presentations for finite soluble groups
7 9
1.6
Acknowledgements
9
.
2 Background theory 2.1 Primitive soluble permutation groups 2.2 Some results from representation theory 2.3 Irreducible cyclic groups over finite fields 2.4 Extraspecial q-groups . 2.5 The irreducible soluble subgroups of G L( n, p) 3 The 3.1 3.2 3.3 3.4 3.5 4
imprimitive soluble subgroups of GL(2,pk) The JS-imprimitives of GL(2,pk) . Miscellaneous results . . . . . . . . The normal subgroups of M contained in the base group The 2-subgroups of M not contained in B The irreducible subgroups of M .
The normaliser of a Singer cycle of prime degree 4.1 The subgroups of M up to M-conjugacy . 4.2 The primitive and cyclic imprimitive subgroups of M
5 The irreducible soluble subgroups of GL(2,pk) 5.1 Miscellaneous results . 5.2 Generating sets for M 3 and M 4 5.2.1 A generating set for M 3 5.2.2 A generating set for M 4
10 10 14 14 16 24 43
43
44 46
49 53
55 55 58 62
63 65 65
68
Contents
VI
5.3
6
7
8
9
The primitive subgroups of M 3 and M 4 5.3.1
The case pk == 3 mod 8
5.3.2 5.3.3
The case pk == 7 mod 8 The case pk == 1 mod 8
5.3.4
The case pk == 5 mod 8
70 71 72 73 74
Some irreducible soluble subgroups of GL(q,pk), q > 2 6.1 The JS-maximals of GL(q,pk) .
75 75
6.2
The imprimitive soluble subgroups of GL(3, 3)
77
6.3 6.4 6.5
The imprimitive soluble subgroups of GL(3, 5) The imprimitive soluble subgroups of GL(5, 3) A generating set for a JS-primitive of GL(3,pk)
80 81
The
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