The Subgroup Structure of the Finite Classical Groups

The Subgroup Structure of the Finite Classical Groups
Title The Subgroup Structure of the Finite Classical Groups PDF eBook
Author Peter B. Kleidman
Publisher Cambridge University Press
Pages 317
Release 1990-04-26
Genre Mathematics
ISBN 052135949X

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With the classification of the finite simple groups complete, much work has gone into the study of maximal subgroups of almost simple groups. In this volume the authors investigate the maximal subgroups of the finite classical groups and present research into these groups as well as proving many new results. In particular, the authors develop a unified treatment of the theory of the 'geometric subgroups' of the classical groups, introduced by Aschbacher, and they answer the questions of maximality and conjugacy and obtain the precise shapes of these groups. Both authors are experts in the field and the book will be of considerable value not only to group theorists, but also to combinatorialists and geometers interested in these techniques and results. Graduate students will find it a very readable introduction to the topic and it will bring them to the very forefront of research in group theory.

The Spread of Almost Simple Classical Groups

The Spread of Almost Simple Classical Groups
Title The Spread of Almost Simple Classical Groups PDF eBook
Author Scott Harper
Publisher Springer Nature
Pages 154
Release 2021-05-25
Genre Mathematics
ISBN 3030741001

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This monograph studies generating sets of almost simple classical groups, by bounding the spread of these groups. Guralnick and Kantor resolved a 1962 question of Steinberg by proving that in a finite simple group, every nontrivial element belongs to a generating pair. Groups with this property are said to be 3/2-generated. Breuer, Guralnick and Kantor conjectured that a finite group is 3/2-generated if and only if every proper quotient is cyclic. We prove a strong version of this conjecture for almost simple classical groups, by bounding the spread of these groups. This involves analysing the automorphisms, fixed point ratios and subgroup structure of almost simple classical groups, so the first half of this monograph is dedicated to these general topics. In particular, we give a general exposition of Shintani descent. This monograph will interest researchers in group generation, but the opening chapters also serve as a general introduction to the almost simple classical groups.

The Maximal Subgroups of the Low-Dimensional Finite Classical Groups

The Maximal Subgroups of the Low-Dimensional Finite Classical Groups
Title The Maximal Subgroups of the Low-Dimensional Finite Classical Groups PDF eBook
Author John N. Bray
Publisher Cambridge University Press
Pages 453
Release 2013-07-25
Genre Mathematics
ISBN 1107276225

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This book classifies the maximal subgroups of the almost simple finite classical groups in dimension up to 12; it also describes the maximal subgroups of the almost simple finite exceptional groups with socle one of Sz(q), G2(q), 2G2(q) or 3D4(q). Theoretical and computational tools are used throughout, with downloadable Magma code provided. The exposition contains a wealth of information on the structure and action of the geometric subgroups of classical groups, but the reader will also encounter methods for analysing the structure and maximality of almost simple subgroups of almost simple groups. Additionally, this book contains detailed information on using Magma to calculate with representations over number fields and finite fields. Featured within are previously unseen results and over 80 tables describing the maximal subgroups, making this volume an essential reference for researchers. It also functions as a graduate-level textbook on finite simple groups, computational group theory and representation theory.

Classical Groups, Derangements and Primes

Classical Groups, Derangements and Primes
Title Classical Groups, Derangements and Primes PDF eBook
Author Timothy C. Burness
Publisher Cambridge University Press
Pages 365
Release 2016-01-15
Genre Mathematics
ISBN 1107629446

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A graduate-level introduction to finite classical groups featuring a comprehensive account of the conjugacy and geometry of elements of prime order.

Linear Algebraic Groups and Finite Groups of Lie Type

Linear Algebraic Groups and Finite Groups of Lie Type
Title Linear Algebraic Groups and Finite Groups of Lie Type PDF eBook
Author Gunter Malle
Publisher Cambridge University Press
Pages 324
Release 2011-09-08
Genre Mathematics
ISBN 113949953X

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Originating from a summer school taught by the authors, this concise treatment includes many of the main results in the area. An introductory chapter describes the fundamental results on linear algebraic groups, culminating in the classification of semisimple groups. The second chapter introduces more specialized topics in the subgroup structure of semisimple groups and describes the classification of the maximal subgroups of the simple algebraic groups. The authors then systematically develop the subgroup structure of finite groups of Lie type as a consequence of the structural results on algebraic groups. This approach will help students to understand the relationship between these two classes of groups. The book covers many topics that are central to the subject, but missing from existing textbooks. The authors provide numerous instructive exercises and examples for those who are learning the subject as well as more advanced topics for research students working in related areas.

The Maximal Subgroups of Classical Algebraic Groups

The Maximal Subgroups of Classical Algebraic Groups
Title The Maximal Subgroups of Classical Algebraic Groups PDF eBook
Author Gary M. Seitz
Publisher American Mathematical Soc.
Pages 294
Release 1987
Genre Linear algebraic groups
ISBN 0821824279

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Let [italic]V be a finite dimensional vector space over an algebraically closed field of characteristic p [greater than] 0 and let G = SL([italic]V), Sp([italic]V), or SO([italic]V). The main result describes all closed, connected, overgroups of [italic]X in SL([italic]V), assuming [italic]X is a closed, connected, irreducible subgroup of G.

The Structure of Groups of Prime Power Order

The Structure of Groups of Prime Power Order
Title The Structure of Groups of Prime Power Order PDF eBook
Author Charles Richard Leedham-Green
Publisher Clarendon Press
Pages 356
Release 2002
Genre Mathematics
ISBN 9780198535485

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An important monograph summarizing the development of a classification system of finite p-groups.