Biggs N.L., White A.T. Permutation Groups and Combinatorial Structures

Cambridge: Cambridge University Press, 1979. — 148 p.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 a set of carefully constructed projects, intended to give the reader a deeper understanding of the subject.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.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).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.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.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.