Permutation Groups and Cartesian Decompositions: (Series Number 449 London Mathematical Society Lecture Note Series)

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.

Table of Contents:
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.

About the Author :
Cheryl E. Praeger is Emeritus Professor at the Centre for the Mathematics of Symmetry and Computation at the University of Western Australia, Perth. She is an Honorary Life Member of the Australian Mathematical Society, and was its first female President. She has authored more than 400 research publications, including five books. Besides holding honorary doctorates awarded by universities in Thailand, Iran, Belgium, Scotland, and Australia, she is also a member of the Order of Australia for her service to mathematics in Australia. Csaba Schneider is Professor in the Maths Department at the Federal University of Minas Gerais, Brazil. He has held research positions at the University of Western Australia, Perth, the Technical University of Braunschweig, the Hungarian Academy of Sciences, and the University of Lisbon. His mathematical interests include finite group theory, the theory of non-associative algebras, and computational algebra.

Review :
'This is a thorough reference book that consists of three parts … In summary, the book is an impressive collection of theorems and their proofs.' Miklós Bóna, MAA Reviews 'One of the most important achievements of this book is building the first formal theory on G-invariant cartesian decompositions; this brings to the fore a better knowledge of the O'Nan–Scott theorem for primitive, quasiprimitive, and innately transitive groups, together with the embeddings among these groups. This is a valuable, useful, and beautiful book.' Pablo Spiga, Mathematical Reviews