Permutation groups and Cartesian decompositions

Description

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.  Read more... Abstract: The theory of permutation groups has a wide range of applications including combinatorics, graph theory, computer science, theoretical physics and molecular chemistry. This book introduces topics that will appeal to students and researchers who require knowledge of permutation group theory for their work and are interested in its applications.  Read more... Content: 1. Introduction Part I. Permutation Groups - Fundamentals: 2. Group actions and permutation groups 3. Minimal normal subgroups of transitive permutation groups 4. Finite direct products of groups 5. Wreath products 6. Twisted wreath products 7. O'Nan-Scott theory and the maximal subgroups of finite alternating and symmetric groups Part II. Innately Transitive Groups - Factorisations and Cartesian Decompositions: 8. Cartesian factorisations 9. Transitive cartesian decompositions for innately transitive groups 10. Intransitive cartesian decompositions Part III. Cartesian Decompositions - Applications: 11. Applications in permutation group theory 12. Applications to graph theory Appendix. Factorisations of simple and characteristically simple groups Glossary References Index.