Finite Groups and Finite Geometries

A 1982 introduction to developments which had taken place in finite group theory related to finite geometries.

T. Tsuzuku (Author), A. Sevenster (Translated by), T. Okuyama (Translated by)

9780521183789, Cambridge University Press

Paperback, published 17 February 2011

342 pages
21.6 x 14 x 1.9 cm, 0.44 kg

The purpose of this 1982 book is to present an introduction to developments which had taken place in finite group theory related to finite geometries. This book is practically self-contained and readers are assumed to have only an elementary knowledge of linear algebra. Among other things, complete descriptions of the following theorems are given in this book; the nilpotency of Frobneius kernels, Galois and Burnside theorems on permutation groups of prime degree, the Omstrom–Wagner theorem on projective planes, and the O'Nan and Ito theorems on characterizations of projective special linear groups. Graduate students and professionals in pure mathematics will continue to find this account of value.

Part I. Introduction: 1. Notation and preliminaries
2. Groups
3. Algebraic structures
4. Vector spaces
5. Geometric structures
Part II. Fundamental Properties of Finite Groups: 1. The Sylow theorems
2. Direct products and semi-direct products
3. Normal series
4. Finite Abelian groups
5. p-groups
6. Groups with operators
7. Group extensions and the theorem of Schur–Zassenhaus
8. Normal ?-complements
9. Normal p-complements
10. Representation of finite groups
11. Frobenius groups
Part III. Fundamental Theory of Permutation Groups: 1. Permutations
2. Transitivity and intransitivity
3. Primitivity and imprimitivity
4. Multiple transitivity
5. Normal subgroups
6. Permutation groups of prime degree
7. Primitive permutation groups
Part IV. Examples - Symmetric Groups and General Linear Groups: 1. Conjugacy classes and composition series of the symmetric and alternating group
2. Conditions for being a symmetric or alternating group
3. Subgroups and automorphism groups of S? and A?
4. Generators and fundamental relations for Sn and An
5. The structure of general semi-linear groups
6. Properties of PSL(V) as a permutation group (dim V ? 3)
7. Symmetric groups and general linear groups of low order
Part V. Finite Projective Geometry: 1. Projective planes and affine planes
2. Higher-dimensional
projective geometry
3. Characterization of projective geometries
Part VI. Finite Groups and Finite Geometries: 1. Designs constructed from 2-transitive groups
2. Characterization of projective transformation
Epilogue
Index.

Subject Areas: Algebra [PBF]