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 MathieuGroups and Steiner Systems. - 6. 1. The Mathieu Groups. - 6. 2. Steiner Systems. - 6. 3. The Extension of AG2(3). - 6. 4. The Mathieu Groups M11and M12. - 6. 5. The Geometry of PG2(4). - 6. 6. The Extension of PG2(4) and the Group M22. - 6. 7. The Mathieu Groups M23and M24. - 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. Language: English