1 First Ideas. - 1. 1 Introduction. - 1. 2 The Definition of a Group. - 1. 3 The General Associative Law. - 1. 4 Further Examples of Groups. - 1. 5 Aims. - Exercises 1. - 2 Multiplication Table Generators Relations Isomorphism. - 2. 1 Multiplication Table. - 2. 2 Multiplication Table for the Dihedral Group D3. - 2. 3 Order of an Element. - 2. 4 The Symmetric Group Sn. - 2. 5 Isomorphism n. - 2. 6 Generators and Relations. - 2. 7 All Possible Groups of Orders 1 2 3 4. - 2. 8 Some Results on Orders of Elements. - Exercises 2. - 3 Subgroups Lagrange's Theorem Cyclic Groups. - 3. 1 Cosets and Lagrange's Theorem. - 3. 2 Some Results on Subgroups. - 3. 3 Generators. - 3. 4 Products of Subsets of Groups. - 3. 5 Cyclic Groups. - 3. 6 Subgroups of S3. - Exercises 3. - 4 Factor Groups Permutation Representations Finite Point Groups. - 4. 1 Normal Subgroups. - 4. 2 Simplicity. - 4. 3 Conjugacy. - 4. 4 Conjugacy Classes. - 4. 5 Homomorphisms. - 4. 6 Permutation Representation of a Group. - 4. 7 Subgroups of Factor Groups. - 4. 8 Factor Groups of Factor Groups. - 4. 9 Groups of Order p2 p prime. - 4. 10 Symmetry and the Orthogonal Group. - 4. 11 Classification of the Finite Rotation Groups. - 4. 12 Examples of Finite Rotation Groups. - 4. 13 Classification of Finite Point Groups of the Second Kind. - 4. 14 Examples of Some of the Finite Point Groups of the Second Kind. - Exercises 4. - 5 Finitely Generated Abelian Groups. - 5. 1 Introduction. - 5. 2 Direct Sum. - 5. 3 Free Abelian Groups. - 5. 4 Structure Theorems for Finitely Generated Abelian Groups. - 5. 5 Uniqueness. - 5. 6 Possible Groups of Order p2. - Exercises 5. - 6 The Sylow Theorems. - 6. 1 Introduction. - 62 Double Cosets. - 6. 3 The Sylow Theorems. - 6. 4 Applications of the Sylow Theorems. - Exercises 6. - 7 Groups of Orders 1 To 15. - 7. 1 Introduction. - 7. 2 Groups of Order 6. - 7. 3 Groups of Order7. - 7. 4 Groups of Order 8. - 7. 5 Groups of Order 9. - 7. 6 Groups of Order 10. - 7. 7 Groups of Order 11. - 7. 8 Groups of Order 12. - 7. 9 Groups of Order 13. - 7. 10 Groups of Order 14. - 7. 11 Groups of Order 15. - 7. 12 Summary. - Exercises 7. - 8 Epilogue. - 8. 1 Introduction. - 8. 2 Construction of Finite Groups. - 8. 3 Solvable and Nilpotent Groups. - 8. 4 The Isomorphism Theorems. - 8. 5 The Schreier-Jordan-Hölder Theorem. - 8. 6 Some Basic Results on Solvable Groups. - Exercises 8. - Miscellaneous Exercises. - Outline Solutions to the Exercises. - Exercises 1. - Exercises 2. - Exercises 3. - Exercises 4. - Exercises 5. - Exercises 6. - Exercises 7. - Exercises 8. - Miscellaneous Exercises. - Further Reading and References. - Intermèdiate. - Advanced. - Sources of Further Problems. - References. - Further Reading and References for Scientists. - General Reference. Language: English