Conjugacy In Finite Classical Groups

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Conjugacy In Finite Classical Groups

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Author : Giovanni De Franceschi
language : en
Publisher: Springer Nature
Release Date : 2025-05-05

Conjugacy In Finite Classical Groups written by Giovanni De Franceschi and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-05-05 with Mathematics categories.

This book provides a comprehensive coverage of the theory of conjugacy in finite classical groups. Given such a classical group G, the three fundamental problems considered are the following: to list a representative for each conjugacy class of G; to describe the centralizer of each representative, by giving its group structure and a generating set; and to solve the conjugacy problem in G—namely, given two elements of G, establish whether they are conjugate, and if so, find a conjugating element. The book presents comprehensive theoretical solutions to all three problems, and uses these solutions to formulate practical algorithms. In parallel to the theoretical work, implementations of these algorithms have been developed in Magma. These form a critical component of various general algorithms in computational group theory—for example, computing character tables and solving conjugacy problems in arbitrary finite groups.

The Subgroup Structure Of The Finite Classical Groups

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Author : Peter B. Kleidman
language : en
Publisher: Cambridge University Press
Release Date : 1990-04-26

The Subgroup Structure Of The Finite Classical Groups written by Peter B. Kleidman and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1990-04-26 with Mathematics categories.

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 Maximal Subgroups Of The Low Dimensional Finite Classical Groups

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Author : John N. Bray
language : en
Publisher: Cambridge University Press
Release Date : 2013-07-25

The Maximal Subgroups Of The Low Dimensional Finite Classical Groups written by John N. Bray and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-07-25 with Mathematics categories.

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.

Centralizers And Conjugacy Classes In Finite Classical Groups

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Author : Giovanni De Franceschi
language : en
Publisher:
Release Date : 2018

Centralizers And Conjugacy Classes In Finite Classical Groups written by Giovanni De Franceschi and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018 with Finite groups categories.

Let G be a classical group defined over a finite field. The aim of this thesis is to obtain a precise and explicit solution to each of the following closely related problems:• List a representative for each conjugacy class of G.• Given x ∈ G, describe the centralizer CG(x) of x in G, by giving its group structure and a generating set.• Given x, y ∈ G, establish whether x and y are conjugate in G and, if they are, find explicit z ∈ G such that z−1xz = y. We present comprehensive theoretical solutions to all three problems. Their solution is often a necessary and vital component of algorithms for computational group theory. Hence we seek explicit solutions which can be employed widely. To achieve this outcome, we use our theoretical solutions to formulate practical algorithms to solve the problems. In parallel to our theoretical work, we have developed in Magma complete implementations of these algorithms.

Classical Groups Derangements And Primes

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Author : Timothy C. Burness
language : en
Publisher: Cambridge University Press
Release Date : 2016-01-15

Classical Groups Derangements And Primes written by Timothy C. Burness and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-01-15 with Mathematics categories.

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

Counting Real Conjugacy Classes In Some Finite Classical Groups

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Author : Elena Amparo
language : en
Publisher:
Release Date : 2018

Counting Real Conjugacy Classes In Some Finite Classical Groups written by Elena Amparo and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018 with Conjugacy classes categories.

An element $g$ in a group $G$ is real if there exists $x\in G$ such that $xgx^{-1}=g^{-1}$. If $g$ is real then all elements in the conjugacy class of $g$ are real. In \cite{GS1} and \cite{GS2}, Gill and Singh showed that the number of real $\mathrm{GL}_n(q)$-conjugacy classes contained in $\mathrm{SL}_n(q)$ equals the number of real $\mathrm{PGL}_n(q)$-conjugacy classes when $q$ is even or $n$ is odd. In this paper, we use generating functions to show that the result is also true for odd $q$. We then follow the methods of \cite{GS1} and \cite{GS2} to count the number of real $\mathrm{U}_n(q)$ conjugacy classes contained in $\mathrm{SU}_n(q)$ and the number of real conjugacy classes in $\mathrm{PGU}_n(q)$, and we show that these are equal to the analogous quantities for $\mathrm{GL}_n(q)$ and $\mathrm{PGL}_n(q)$. Thus, we show that these four sets of conjugacy classes have equal size for all $n,q$.

Conjugacy Classes In Semisimple Algebraic Groups

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Author : James E. Humphreys
language : en
Publisher: American Mathematical Soc.
Release Date : 1995

Conjugacy Classes In Semisimple Algebraic Groups written by James E. Humphreys and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1995 with Education categories.

Provides a useful exposition of results on the structure of semisimple algebraic groups over an arbitrary algebraically closed field. After the fundamental work of Borel and Chevalley in the 1950s and 1960s, further results were obtained over the next thirty years on conjugacy classes and centralizers of elements of such groups.

A Course In Finite Group Representation Theory

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Author : Peter Webb
language : en
Publisher: Cambridge University Press
Release Date : 2016-08-19

A Course In Finite Group Representation Theory written by Peter Webb and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-08-19 with Mathematics categories.

This graduate-level text provides a thorough grounding in the representation theory of finite groups over fields and rings. The book provides a balanced and comprehensive account of the subject, detailing the methods needed to analyze representations that arise in many areas of mathematics. Key topics include the construction and use of character tables, the role of induction and restriction, projective and simple modules for group algebras, indecomposable representations, Brauer characters, and block theory. This classroom-tested text provides motivation through a large number of worked examples, with exercises at the end of each chapter that test the reader's knowledge, provide further examples and practice, and include results not proven in the text. Prerequisites include a graduate course in abstract algebra, and familiarity with the properties of groups, rings, field extensions, and linear algebra.

Groups Combinatorics Geometry

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Author : A. A. Ivanov
language : en
Publisher: World Scientific
Release Date : 2003

Groups Combinatorics Geometry written by A. A. Ivanov and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003 with Mathematics categories.

"This book contains the proceedings of the L.M.S. Durham Symposium on Groups, Geometry and Combinatorics, July 16-26, 2001"--P. v.

Beauville Surfaces And Groups

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Author : Ingrid Bauer
language : en
Publisher: Springer
Release Date : 2015-04-14

Beauville Surfaces And Groups written by Ingrid Bauer and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-04-14 with Mathematics categories.

This collection of surveys and research articles explores a fascinating class of varieties: Beauville surfaces. It is the first time that these objects are discussed from the points of view of algebraic geometry as well as group theory. The book also includes various open problems and conjectures related to these surfaces. Beauville surfaces are a class of rigid regular surfaces of general type, which can be described in a purely algebraic combinatoric way. They play an important role in different fields of mathematics like algebraic geometry, group theory and number theory. The notion of Beauville surface was introduced by Fabrizio Catanese in 2000 and after the first systematic study of these surfaces by Ingrid Bauer, Fabrizio Catanese and Fritz Grunewald, there has been an increasing interest in the subject. These proceedings reflect the topics of the lectures presented during the workshop ‘Beauville surfaces and groups 2012’, held at Newcastle University, UK in June 2012. This conference brought together, for the first time, experts of different fields of mathematics interested in Beauville surfaces.