Permutation groups: theory and applications. (Q2810316)

This book is based on a course of ten lectures given by the author at a summer school on coherent configurations, permutations groups and applications in algebraic graph theory at Nový Smokovec in 2014. The author's aim is to provide a broad survey of topics on (generally finite) permutation groups in a combinatorial setting; there are many examples and exercises, but very few proofs.NEWLINENEWLINE The following outline of the chapters describes the book: (1) and (2) (Basic examples and definitions) automorphism group of the Petersen graph, action of the symmetric group on pairs of points, regular permutation groups, stabilizing series and group orders, a brief statement of the classification of finite simple groups; (3) (Invariant relations) relations and their relationship to transitivity and primitivity; (4) (Primitive groups) normal subgroups of primitive groups, structure of primitive groups (with reference to the O'Nan-Scott theorem), dichotomy between the alternating and symmetric groups and their smaller primitive subgroups; (5) (Multiply transitive groups) Steiner systems, \(k\)-homogeneous groups; (6) (Representation theory and character theory) a brief outline of ordinary representations; (7) (Representation theory for permutation groups) characters of permutation groups and the character table for \(S_4\), permutation groups of prime degree; (8) (Centralizer algebra) Basic properties, \(S\)-rings, \(B\)-groups; (9) (Hamming and Johnson graphs, groups and schemes); (10) (Permutation groups and maps) Frobenius groups, maps of connected graphs and monodromy groups, regular embeddings of complete graphs. The book concludes with a brief summary of notation and fundamental definitions for permutation groups, hints and solutions to the exercises, and a bibliography for references and further reading.NEWLINENEWLINE It is assumed that the reader has familiarity with standard undergraduate courses in linear algebra, group theory, fields, modules and algebraic numbers and an introduction to graph theory. This book is clearly written and could serve as the text for a course or for independent study.