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The study of permutation groups has always been closely associated with that of highly symmetric structures. The objects considered here are countably infinite, but have only finitely many different substructures of any given finite size. They are precisely those structures which are determined by first-order logical axioms together with the assumption of countability. This book concerns such structures, their substructures and their automorphism groups. A wide range of techniques are used: group theory, combinatorics, Baire category and measure among them. The book arose from lectures given at a research symposium and retains their informal style, whilst including as well many recent results from a variety of sources. It concludes with exercises and unsolved research problems.

Permutation groups.

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Permutation groups, their fundamental theory and applications are discussed in this introductory book. It focuses on those groups that are most useful for studying symmetric structures such as graphs, codes and designs. Modern treatments of the O'Nan-Scott theory are presented not only for primitive permutation groups but also for the larger families of quasiprimitive and innately transitive groups, including several classes of infinite permutation groups. Their precision is sharpened by the introduction of a cartesian decomposition concept. This facilitates reduction arguments for primitive groups analogous to those, using orbits and partitions, that reduce problems about general permutation groups to primitive groups. The results are particularly powerful for finite groups, where the finite simple group classification is invoked. Applications are given in algebra and combinatorics to group actions that preserve cartesian product structures. Students and researchers with an interest in mathematical symmetry will find the book enjoyable and useful.

Permutation groups.

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Permutation groups. --- Group theory.

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The study of permutation groups has always been closely associated with that of highly symmetric structures. The objects considered here are countably infinite, but have only finitely many different substructures of any given finite size. They are precisely those structures which are determined by first-order logical axioms together with the assumption of countability. This book concerns such structures, their substructures and their automorphism groups. A wide range of techniques are used: group theory, combinatorics, Baire category and measure among them. The book arose from lectures given at a research symposium and retains their informal style, whilst including as well many recent results from a variety of sources. It concludes with exercises and unsolved research problems.

Permutation groups. --- Substitution groups --- Group theory

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Finite groups. --- Mathieu groups. --- Permutation groups.