Permutation Groups and Cartesian Decompositions

Concise introduction to permutation groups, focusing on invariant cartesian decompositions and applications in algebra and combinatorics.

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.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.