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The subject of this book is the action of permutation groups on sets associated with combinatorial structures. Each chapter deals with a particular structure: groups, geometries, designs, graphs and maps respectively. A unifying theme for the first four chapters is the construction of finite simple groups. In the fifth chapter, a theory of maps on orientable surfaces is developed within a combinatorial framework. This simplifies and extends the existing literature in the field. The book is designed both as a course text and as a reference book for advanced undergraduate and graduate students. A feature is the set of carefully constructed projects, intended to give the reader a deeper understanding of the subject.
Introduction
Note on the Projects
Preliminary definitions
Counting principles
Transitivity
Applications to group theory
Extensions of multiply transitive groups
Primitivity
Regular normal subgroups
Project: Proof of Sylow's theorem
Project: Some multiply transitive groups
Notes and references
Introduction
Finite fields
Finite vector spaces
The structure of GL(V) and SL(V)
Projective spaces and their groups
More about projective spaces
The classical simple groups
Project: Near-fields and sharply -transitive groups
Project: A unitary polarity in PG(2, 9)
Notes and references
Four fundamental problems
Designs
Symmetric designs
Automorphisms of designs
Extensions of designs
Mathieu groups and associated designs
Project: Hadamard matrices and designs
Project: Uniqueness of the 3 - (22, 6, 1) design
Notes and references
Permutation groups and graphs
Automorphisms of graphs
Rank groups and the associated graphs
Feasibility conditions for strongly regular graphs
The Higman-Sims group
Project: Some graphs and their automorphism groups
Project: Strongly regular graphs and biplanes
Notes and references
Maps and surfaces
Automorphisms of maps
Cayley graphs and Cayley maps
Complete maps and a theorem of Frobenius
Symmetrical maps
Project: Generalized Cayley maps
Project: Paley maps
Project: Symmetrical Cayley maps
Notes and references
Index