Martin W. Liebeck's Books

Intends to complete the determination of the maximal subgroups of positive dimension in simple algebraic groups of exceptional type over algebraically closed fields. This title follows work of Dynkin, who solved the problem in characteristic zero, and Seitz who did likewise over fields whose characteristic is not too small.

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.

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.

Addresses the classical problem of determining finite primitive permutation groups G with a regular subgroup B.

This book gives a proof of Cherlin’s conjecture for finite binary primitive permutation groups. Motivated by the part of model theory concerned with Lachlan’s theory of finite homogeneous relational structures, this conjecture proposes a classification of those finite primitive permutation groups that have relational complexity equal to 2. The first part gives a full introduction to Cherlin’s conjecture, including all the key ideas that have been used in the literature to prove some of its special cases. The second part completes the proof by dealing with primitive permutation groups that are almost simple with socle a group of Lie type. A great deal of material concerning properties of primitive permutation groups and almost simple groups is included, and new ideas are introduced. Addressing a hot topic which cuts across the disciplines of group theory, model theory and logic, this book will be of interest to a wide range of readers. It will be particularly useful for graduate students and researchers who need to work with simple groups of Lie type.

The theory of simple algebraic groups is important in many areas of mathematics. The authors of this book investigate the subgroups of certain types of simple algebraic groups and obtain a complete description of all those subgroups which are themselves simple. This description is particularly useful in understanding centralizers of subgroups and restrictions of representations.

This volume contains a collection of papers on the subject of the classification of finite simple groups.

Introducing the representation theory of finite groups, this second edition has been revised and updated. The theory is developed in terms of modules with considerable emphasis placed upon constructing characters.

The main result describes completely the maximal factorizations of all the finite simple groups and their automorphism groups. As a consequence, a classification of the maximal subgroups of the finite alternating and symmetric groups is obtained.

Over the past 20 years, the theory of groups in particular simplegroups, finite and algebraic has influenced a number of diverseareas of mathematics. Such areas include topics where groups have beentraditionally applied, such as algebraic combinatorics, finitegeometries, Galois theory and permutation groups, as well as severalmore recent developments.

Book's by Martin W. Liebeck

The Maximal Subgroups of Positive Dimension in Exceptional Algebraic Groups

The Maximal Subgroups of Positive Dimension in Exceptional Algebraic Groups

The Subgroup Structure of the Finite Classical Groups

The Subgroup Structure of the Finite Classical Groups

Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras

Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras

Regular Subgroups of Primitive Permutation Groups

Regular Subgroups of Primitive Permutation Groups

Multiplicity-free Representations of Algebraic Groups

Multiplicity-free Representations of Algebraic Groups

Cherlin’s Conjecture for Finite Primitive Binary Permutation Groups

Cherlin’s Conjecture for Finite Primitive Binary Permutation Groups

Reductive Subgroups of Exceptional Algebraic Groups

Reductive Subgroups of Exceptional Algebraic Groups

Groups, Combinatorics and Geometry

Groups, Combinatorics and Geometry

Representations and Characters of Groups

Representations and Characters of Groups

The Maximal Factorizations of the Finite Simple Groups and Their Automorphism Groups

The Maximal Factorizations of the Finite Simple Groups and Their Automorphism Groups

Groups Combinatorics & Geometry

Groups Combinatorics & Geometry

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