Author | Gardiner, Cyril F. author |
---|---|
Title | A First Course in Group Theory [electronic resource] / by Cyril F. Gardiner |
Imprint | New York, NY : Springer New York, 1980 |
Connect to | http://dx.doi.org/10.1007/978-1-4613-8117-4 |
Descript | X, 228 p. online resource |
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, Lagrangeโs Theorem, Cyclic Groups -- 3.1 Cosets and Lagrangeโs 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-Hรถlder 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 -- Intermรจdiate -- Advanced -- Sources of Further Problems -- References -- Further Reading and References for Scientists -- General Reference