Finite Groups and Finite Geometries

Table of Contents

• 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 π-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 SΩ and AΩ
• 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.
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