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Permutation groups are one of the oldest topics in algebra. However, their study has recently been revolutionised by new developments, particularly the classification of finite simple groups, but also relations with logic and combinatorics, and importantly, computer algebra systems have been introduced that can deal with large permutation groups. This book gives a summary of these developments, including an introduction to relevant computer algebra systems, sketch proofs of major theorems, and many examples of applying the classification of finite simple groups. It is aimed at beginning graduate students and experts in other areas, and grew from a short course at the EIDMA institute in Eindhoven.

Permutation groups --- Permutation groups. --- Groupes de permutations --- Substitution groups --- Group theory

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Group theory --- Permutation groups. --- Groupes de permutations --- Permutation groups --- 512.542.7 --- Statistics --- -Statistical analysis --- Statistical data --- Statistical methods --- Statistical science --- Mathematics --- Econometrics --- Problems, exercises, etc --- -Permutation groups --- 512.542.7 Permutation groups --- -512.542.7 Permutation groups --- Statistical analysis --- Substitution groups

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Ordered algebraic structures --- 512 --- Algebra --- 512 Algebra --- Permutation groups. --- Permutation groups --- Groupes de permutations --- Group theory --- Representations of groups. --- Representations of groups --- Représentations de groupes --- Représentations de groupes --- Groupes finis --- Groupes (algebre) --- Representation

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The subject of this book is the action of permutation groups on sets associated with combinatorial structures. Each chapter deals with a particular structure: groups, geometries, designs, graphs and maps respectively. A unifying theme for the first four chapters is the construction of finite simple groups. In the fifth chapter, a theory of maps on orientable surfaces is developed within a combinatorial framework. This simplifies and extends the existing literature in the field. The book is designed both as a course text and as a reference book for advanced undergraduate and graduate students. A feature is the set of carefully constructed projects, intended to give the reader a deeper understanding of the subject.

Combinatorial analysis. --- Permutation groups. --- Ordered algebraic structures --- Discrete mathematics --- 512.54 --- Combinatorial analysis --- Permutation groups --- Substitution groups --- Group theory --- Combinatorics --- Algebra --- Mathematical analysis --- 512.54 Groups. Group theory --- Groups. Group theory --- Groupes de permutations --- Analyse combinatoire

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This is the first-ever book on computational group theory. It provides extensive and up-to-date coverage of the fundamental algorithms for permutation groups with reference to aspects of combinatorial group theory, soluble groups, and p-groups where appropriate. The book begins with a constructive introduction to group theory and algorithms for computing with small groups, followed by a gradual discussion of the basic ideas of Sims for computing with very large permutation groups, and concludes with algorithms that use group homomorphisms, as in the computation of Sylowsubgroups. No background in group theory is assumed. The emphasis is on the details of the data structures and implementation which makes the algorithms effective when applied to realistic problems. The algorithms are developed hand-in-hand with the theoretical and practical justification.All algorithms are clearly described, examples are given, exercises reinforce understanding, and detailed bibliographical remarks explain the history and context of the work. Much of the later material on homomorphisms, Sylow subgroups, and soluble permutation groups is new.

Group theory --- Computer science --- Algorithmes --- Algorithms --- Algoritmen --- Permutatiegroepen --- Permutation [Groupes de ] --- Permutation groups --- Permutation groups. --- Algorithms. --- Groupes de permutations --- 681.3*I12 --- Substitution groups --- Algorism --- Algebra --- Arithmetic --- Algorithms: algebraic algorithms; nonalgebraic algorithms; analysis of algorithms (Algebraic manipulation; computing methodologies) --- Foundations --- 681.3*I12 Algorithms: algebraic algorithms; nonalgebraic algorithms; analysis of algorithms (Algebraic manipulation; computing methodologies) --- Information theory. --- Group theory. --- Computer software. --- Combinatorics. --- Theory of Computation. --- Group Theory and Generalizations. --- Discrete Mathematics. --- Symbolic and Algebraic Manipulation. --- Algorithm Analysis and Problem Complexity. --- Data processing. --- Combinatorics --- Mathematical analysis --- Software, Computer --- Computer systems --- Groups, Theory of --- Substitutions (Mathematics) --- Communication theory --- Communication --- Cybernetics