## A Course in Group TheoryThe classification of the finite simple groups is one of the major intellectual achievements of this century, but it remains almost completely unknown outside of the mathematics community. This introduction to group theory is also an attempt to make this important work better known. Emphasizing classification themes throughout, the book gives a clear and comprehensive introduction to groups and covers all topics likely to be encountered in an undergraduate course. Introductory chapters explain the concepts of group, subgroup and normal subgroup, and quotient group. The homomorphism and isomorphism theorems are explained, along with an introduction to G-sets. Subsequent chapters deal with finite abelian groups, the Jordan-Holder theorem, soluble groups, p-groups, and group extensions. The numerous worked examples and exercises in this excellent and self-contained introduction will also encourage undergraduates (and first year graduates) to further study. |

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### Contents

Definitions and examples | 1 |

Maps and relations on sets | 8 |

Elementary consequences of the definitions | 18 |

Subgroups | 30 |

Cosets and Lagranges Theorem | 38 |

Errorcorrecting codes | 49 |

Normal subgroups and quotient groups | 59 |

The Homomorphism Theorem | 68 |

Composition factors and chief factors | 137 |

Soluble groups | 146 |

Examples of soluble groups | 155 |

Semidirect products and wreath products | 163 |

Extensions | 174 |

Central and cyclic extensions | 183 |

Groups with at most 31 elements | 192 |

The projective special linear groups | 202 |

Permutations | 77 |

The OrbitStabiliser Theorem | 89 |

The Sylow Theorems | 98 |

Applications of Sylow theory | 106 |

Direct products | 112 |

The classification of finite abelian groups | 120 |

The JordanHolder Theorem | 128 |

The Mathieu groups | 213 |

The classification of finite simple groups | 222 |

A Prerequisites from number theory and linear | 234 |

Solutions to exercises | 243 |

275 | |

### Common terms and phrases

Aut(G axiom bijection centraliser chief series codeword commute composition series congruent conjugacy classes conjugate consider contains Corollary cycle cyclic group define Definition denote dihedral group divisor element g element of G element of order Example extension finite group finite simple groups follows by Proposition G and H g in G G is abelian G of order G-set group G group of order homomorphism identity element integer internal direct product inverse Jordan-Holder Theorem Lagrange's Theorem left cosets Let G matrix modulo multiplication nilpotent nilpotent group non-abelian group non-identity element non-zero normal series normal subgroup number of elements number of Sylow p-group permutation positive integer prime quaternion group quotient group result right coset semidirect product soluble groups stabiliser subgroup H subgroup of G subgroup of order subset Summary for Chapter Suppose that G surjective Sylow p-subgroup symmetric group unique Sylow vector wreath product