## The Theory of GroupsUseful, well-written graduate level text designed to acquaint the reader with group-theoretic methods and to demonstrate their usefulness as tools in the solution of mathematical and physical problems. Covers such subjects as axioms, the calculus of complexes, homomorphic mapping, p-group theory and more. Many proofs are shorter and more transparent than older ones. |

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### Contents

ELEMENTS OF GROUP THEORY | 1 |

Investigation of Axioms | 9 |

Cyclic Groups | 15 |

The Concept of Normal Subgroup | 23 |

THE CONCEPT OF HOMOMORPHV | 35 |

Operators and Operator Homomorphies | 44 |

Normal Chains and Normal Series | 57 |

Commutator Groups and Commutator Forms | 78 |

Theorems on Sylow pGroups | 138 |

On the Enumeration Theorems of the Theory of pGroups | 153 |

Hamiltonian Groups | 159 |

The Theorems of Burnside and Grim | 169 |

The Principal Ideal Theorem | 176 |

B Structure Theory and Direct Products A Treatment | 188 |

Free Products and Groups Given by a Set of Generators | 217 |

Further Exercises for Chap Ill | 230 |

On the Groups of an Algebra | 84 |

THE STRUCTURE AND CONSTRUCTION | 109 |

Abelian Groups | 117 |

On the Order Ideal | 123 |

Extension Theory | 124 |

Extensions with Abelian Factor Group | 131 |

E Further Exercises for Chap IV 5 | 238 |

G A Theorem of Wielandt An Addendum to Chap IV | 245 |

H Further Exercises for Chap V I | 252 |

259 | |

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### Common terms and phrases

abelian group algebra alternating permutation apply belongs binary relation called complemented complex composition series congruence relation conjugate contained Conversely corresponding cyclic group DEFINITION denoted direct product division ring divisor equal equation equivalent Exercise factor group factor lattices factor system finite group finite number follows given group of order hence homomorphism implies induces an isomorphism induction hypothesis intersection left cosets letters linear matrix modular lattice module Moreover morphism natural numbers nilpotent group non-abelian normal chain normal operator number of elements obtain permutation group poset product over H projective Proof prove quasi-ring quaternion group rational integers relatively prime Remak decomposition representation representative residue class rotations rule satisfying semi-group semi-ring Show single-valued mapping smallest normal subgroup solvable subgroup of index subgroup of order sublattice subnormal subgroups subset Sylow p-group system of imprimitivity theorem theory transformations transitive uniquely determined unit element vector zero element