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Permutation Groups


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Table of Contents

1. The Basic Ideas.- 1.1. Symmetry.- 1.2. Symmetric Groups.- 1.3. Group Actions.- 1.4. Orbits and Stabilizers.- 1.5. Blocks and Primitivity.- 1.6. Permutation Representations and Normal Subgroups.- 1.7. Orbits and Fixed Points.- 1.8. Some Examples from the Early History of Permutation Groups.- 1.9. Notes.- 2. Examples and Constructions.- 2.1. Actions on k-tuples and Subsets.- 2.2. Automorphism Groups of Algebraic Structures.- 2.3. Graphs.- 2.4. Relations.- 2.5. Semidirect Products.- 2.6. Wreath Products and Imprimitive Groups.- 2.7. Primitive Wreath Products.- 2.8. Affine and Projective Groups.- 2.9. The Transitive Groups of Degree at Most 7.- 2.10. Notes.- 3. The Action of a Permutation Group.- 3.1. Introduction.- 3.2. Orbits of the Stabilizer.- 3.3. Minimal Degree and Bases.- 3.4. Frobenius Groups.- 3.5. Permutation Groups Which Contain a Regular Subgroup.- 3.6. Computing in Permutation Groups.- 3.7. Notes.- 4. The Structure of a Primitive Group.- 4.1. Introduction.- 4.2. Centralizers and Normalizers in the Symmetric Group.- 4.3. The Socle.- 4.4. Subnormal Subgroups and Primitive Groups.- 4.5. Constructions of Primitive Groups with Nonregular Socles.- 4.6. Finite Primitive Groups with Nonregular Socles.- 4.7. Primitive Groups with Regular Socles.- 4.8. Applications of the O'Nan-Scott Theorem.- 4.9. Notes.- 5. Bounds on Orders of Permutation Groups.- 5.1. Orders of Elements.- 5.2. Subgroups of Small Index in Finite Alternating and Symmetric Groups.- 5.3. The Order of a Simply Primitive Group.- 5.4. The Minimal Degree of a 2-transitive Group.- 5.5. The Alternating Group as a Section of a Permutation Group.- 5.6. Bases and Orders of 2-transitive Groups.- 5.7. The Alternating Group as a Section of a Linear Group.- 5.8. Small Subgroups of Sn.- 5.9. Notes.- 6. The Mathieu Groups and Steiner Systems.- 6.1. The Mathieu Groups.- 6.2. Steiner Systems.- 6.3. The Extension of AG2 (3).- 6.4. The Mathieu Groups M 11 and M12.- 6.5. The Geometry of PG 2 (4).- 6.6. The Extension of PG 2 (4) and the Group M 22.- 6.7. The Mathieu Groups M 23 and M 24.- 6.8. The Geometry of W24.- 6.9. Notes.- 7. Multiply Transitive Groups.- 7.1. Introduction.- 7.2. Normal Subgroups.- 7.3. Limits to Multiple Transitivity.- 7.4. Jordan Groups.- 7.5. Transitive Extensions.- 7.6. Sharply k-transitive Groups.- 7.7. The Finite 2-transitive Groups.- 7.8. Notes.- 8. The Structure of the Symmetric Groups.- 8.1. The Normal Structure of Sym(?).- 8.2. The Automorphisms of Sym(?).- 8.3. Subgroups of F Sym(?).- 8.4. Subgroups of Small Index in Sym(?).- 8.5. Maximal Subgroups of the Symmetric Groups.- 8.6. Notes.- 9. Examples and Applications of Infinite Permutation Groups.- 9.1. The Construction of a Finitely Generated Infinite p-group.- 9.2. Groups Acting on Trees.- 9.3. Highly Transitive Free Subgroups of the Symmetric Group.- 9.4. Homogeneous Groups.- 9.5. Automorphisms of Relational Structures.- 9.6. The Universal Graph.- 9.7. Notes.- Appendix A. Classification of Finite Simple Groups.- Appendix B. The Primitive Permutation Groups of Degree Less than 1000.- References.

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