Permutation Groups and Cartesian Decompositions

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

Cheryl E. Praeger (Author), Csaba Schneider (Author)

9780521675062, Cambridge University Press

Paperback, published 3 May 2018

334 pages, 2 b/w illus. 14 tables
22.7 x 15.1 x 2 cm, 0.5 kg

'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

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.

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.

Subject Areas: Combinatorics & graph theory [PBV], Groups & group theory [PBG], Algebra [PBF]