Centralizers And Conjugacy Classes In Finite Classical Groups
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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.
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.
Conjugacy Classes And Centralizers Of Semisimple Elements In Finite Groups Of Lie Type
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Author : D. I. Deriziotis
language : en
Publisher:
Release Date : 1984
Conjugacy Classes And Centralizers Of Semisimple Elements In Finite Groups Of Lie Type written by D. I. Deriziotis and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1984 with Finite groups categories.
Unipotent And Nilpotent Classes In Simple Algebraic Groups And Lie Algebras
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Author : Martin W. Liebeck
language : en
Publisher: American Mathematical Soc.
Release Date : 2012-01-25
Unipotent And Nilpotent Classes In Simple Algebraic Groups And Lie Algebras written by Martin W. Liebeck 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 2012-01-25 with Mathematics categories.
This book concerns the theory of unipotent elements in simple algebraic groups over algebraically closed or finite fields, and nilpotent elements in the corresponding simple Lie algebras. These topics have been an important area of study for decades, with applications to representation theory, character theory, the subgroup structure of algebraic groups and finite groups, and the classification of the finite simple groups. The main focus is on obtaining full information on class representatives and centralizers of unipotent and nilpotent elements. Although there is a substantial literature on this topic, this book is the first single source where such information is presented completely in all characteristics. In addition, many of the results are new--for example, those concerning centralizers of nilpotent elements in small characteristics. Indeed, the whole approach, while using some ideas from the literature, is novel, and yields many new general and specific facts concerning the structure and embeddings of centralizers.
Linear Algebraic Groups And Finite Groups Of Lie Type
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Author : Gunter Malle
language : en
Publisher: Cambridge University Press
Release Date : 2011-09-08
Linear Algebraic Groups And Finite Groups Of Lie Type written by Gunter Malle 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 2011-09-08 with Mathematics categories.
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.
Finite And Locally Finite Groups
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Author : B. Hartley
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Finite And Locally Finite Groups written by B. Hartley and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
This volume contains the proceedings of the NATO Advanced Study Institute on Finite and Locally Finite Groups held in Istanbul, Turkey, 14-27 August 1994, at which there were about 90 participants from some 16 different countries. The ASI received generous financial support from the Scientific Affairs Division of NATO. INTRODUCTION A locally finite group is a group in which every finite set of elements is contained in a finite subgroup. The study of locally finite groups began with Schur's result that a periodic linear group is, in fact, locally finite. The simple locally finite groups are of particular interest. In view of the classification of the finite simple groups and advances in representation theory, it is natural to pursue classification theorems for simple locally finite groups. This was one of the central themes of the Istanbul conference and significant progress is reported herein. The theory of simple locally finite groups intersects many areas of group theory and representation theory, so this served as a focus for several articles in the volume. Every simple locally finite group has what is known as a Kegel cover. This is a collection of pairs {(G , Ni) liE I}, where I is an index set, each group Gi is finite, i Ni
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.
Seminar On Algebraic Groups And Related Finite Groups
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Author : Armand Borel
language : en
Publisher: Springer
Release Date : 2006-11-14
Seminar On Algebraic Groups And Related Finite Groups written by Armand Borel and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-14 with Mathematics categories.
The Classification Of The Finite Simple Groups Number 8
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Author : Daniel Gorenstein
language : en
Publisher: American Mathematical Soc.
Release Date : 2018-12-12
The Classification Of The Finite Simple Groups Number 8 written by Daniel Gorenstein 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 2018-12-12 with Mathematics categories.
This book completes a trilogy (Numbers 5, 7, and 8) of the series The Classification of the Finite Simple Groups treating the generic case of the classification of the finite simple groups. In conjunction with Numbers 4 and 6, it allows us to reach a major milestone in our series—the completion of the proof of the following theorem: Theorem O: Let G be a finite simple group of odd type, all of whose proper simple sections are known simple groups. Then either G is an alternating group or G is a finite group of Lie type defined over a field of odd order or G is one of six sporadic simple groups. Put another way, Theorem O asserts that any minimal counterexample to the classification of the finite simple groups must be of even type. The work of Aschbacher and Smith shows that a minimal counterexample is not of quasithin even type, while this volume shows that a minimal counterexample cannot be of generic even type, modulo the treatment of certain intermediate configurations of even type which will be ruled out in the next volume of our series.
The Arcata Conference On Representations Of Finite Groups Part 2
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Author : Paul Fong
language : en
Publisher: American Mathematical Soc.
Release Date : 1987
The Arcata Conference On Representations Of Finite Groups Part 2 written by Paul Fong 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 1987 with Mathematics categories.
The papers in these proceedings of the 1986 Arcata Summer Institute bear witness to the extraordinarily vital and intense research in the representation theory of finite groups. The confluence of diverse mathematical disciplines has brought forth work of great scope and depth. Particularly striking is the influence of algebraic geometry and cohomology theory in the modular representation theory and the character theory of reductive groups over finite fields, and in the general modular representation theory of finite groups. The continuing developments in block theory and the general character theory of finite groups is noteworthy. The expository and research aspects of the Summer Institute are well represented by these papers.