The Spread of Almost Simple Classical Groups
The Spread of Almost Simple Classical Groups

Author: Scott Harper

Publisher: Springer Nature

Published: 2021-05-25

Total Pages: 154

ISBN-13: 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.

Classical Groups, Derangements and Primes
Classical Groups, Derangements and Primes

Author: Timothy C. Burness

Publisher: Cambridge University Press

Published: 2016-01-15

Total Pages: 365

ISBN-13: 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.

Groups St Andrews 2017 in Birmingham
Groups St Andrews 2017 in Birmingham

Author: C. M. Campbell

Publisher: Cambridge University Press

Published: 2019-04-11

Total Pages: 510

ISBN-13: 110872874X

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These proceedings of 'Groups St Andrews 2017' provide a snapshot of the state-of-the-art in contemporary group theory.

Surveys in Combinatorics 2021
Surveys in Combinatorics 2021

Author: Konrad K. Dabrowski

Publisher: Cambridge University Press

Published: 2021-06-24

Total Pages: 380

ISBN-13: 1009041819

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This volume contains nine survey articles based on plenary lectures given at the 28th British Combinatorial Conference, hosted online by Durham University in July 2021. This biennial conference is a well-established international event, attracting speakers from around the world. Written by some of the foremost researchers in the field, these surveys provide up-to-date overviews of several areas of contemporary interest in combinatorics. Topics discussed include maximal subgroups of finite simple groups, Hasse–Weil type theorems and relevant classes of polynomial functions, the partition complex, the graph isomorphism problem, and Borel combinatorics. Representing a snapshot of current developments in combinatorics, this book will be of interest to researchers and graduate students in mathematics and theoretical computer science.

Surveys in Combinatorics 2005
Surveys in Combinatorics 2005

Author: Bridget S. Webb

Publisher: Cambridge University Press

Published: 2005-07-21

Total Pages: 270

ISBN-13: 9780521615235

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This volume provides an up-to-date overview of current research across combinatorics,.

Expansion in Finite Simple Groups of Lie Type
Expansion in Finite Simple Groups of Lie Type

Author: Terence Tao

Publisher: American Mathematical Soc.

Published: 2015-04-16

Total Pages: 319

ISBN-13: 1470421968

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Expander graphs are an important tool in theoretical computer science, geometric group theory, probability, and number theory. Furthermore, the techniques used to rigorously establish the expansion property of a graph draw from such diverse areas of mathematics as representation theory, algebraic geometry, and arithmetic combinatorics. This text focuses on the latter topic in the important case of Cayley graphs on finite groups of Lie type, developing tools such as Kazhdan's property (T), quasirandomness, product estimates, escape from subvarieties, and the Balog-Szemerédi-Gowers lemma. Applications to the affine sieve of Bourgain, Gamburd, and Sarnak are also given. The material is largely self-contained, with additional sections on the general theory of expanders, spectral theory, Lie theory, and the Lang-Weil bound, as well as numerous exercises and other optional material.

Groups St Andrews 2013
Groups St Andrews 2013

Author: C. M. Campbell

Publisher: Cambridge University Press

Published: 2015-10-22

Total Pages: 503

ISBN-13: 1316467910

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Every four years, leading researchers gather to survey the latest developments in all aspects of group theory. Since 1981, the proceedings of those meetings have provided a regular snapshot of the state of the art in group theory and helped to shape the direction of research in the field. This volume contains selected papers from the 2013 meeting held in St Andrews. It begins with major articles from each of the four main speakers: Emmanuel Breuillard (Paris-Sud), Martin Liebeck (Imperial College London), Alan Reid (Texas) and Karen Vogtmann (Cornell). These are followed by, in alphabetical order, survey articles contributed by other conference participants, which cover a wide spectrum of modern group theory.

The Subgroup Structure of the Finite Classical Groups
The Subgroup Structure of the Finite Classical Groups

Author: Peter B. Kleidman

Publisher: Cambridge University Press

Published: 1990-04-26

Total Pages: 317

ISBN-13: 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.