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.- 6,2 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 Order 7.- 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-Holder 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.- Intermediate.- Advanced.- Sources of Further Problems.- References.- Further Reading and References for Scientists.- General Reference.
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