## Theory of Groups of Finite OrderAfter introducing permutation notation and defining group, the author discusses the simpler properties of group that are independent of their modes of representation; composition-series of groups; isomorphism of a group with itself; Abelian groups; groups whose orders are the powers of primes; Sylow's theorem; more. 18 illustrations. A classic introduction. |

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### Contents

CHAPTER I | 1 |

Transpositions representation of a permutation as | 11 |

Various modes of representing groups | 24 |

CHAPTER IV | 46 |

ON THE COMPOSITIONSERIES OF A GROUP | 64 |

CHAPTER VI | 81 |

CHAPTER VIII | 119 |

CHAPTER IX | 149 |

Set of groupcharacteristics in conjugate repre | 293 |

CHAPTER XVIII | 372 |

the generating operations are of finite order 389394 | 389 |

CHAPTER XIX | 402 |

CHAPTER XX | 428 |

Examples 457459 | 460 |

On the completely reduced form of a group | 470 |

NOTE F On groups of finite order which are simply iso | 476 |

Conjugate permutations are similar selfconjugate | 173 |

CHAPTER XII | 231 |

CHAPTER XIII | 243 |

Characteristic equation of a substitution charac | 250 |

CHAPTER XIV | 269 |

CHAPTER XV | 280 |

NOTE H On the group of the twentyseven lines on | 485 |

NOTE J On conditions for the finiteness of the order of | 491 |

On the groupcharacteristics of the fractional linear | 499 |

507 | |

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### Common terms and phrases

Abelian group algebraic alternating group circular permutation coefficients common sub-group congruence conjugate operations conjugate sub-groups contained self-conjugately cyclical group cyclical sub-group defined denoted direct product equation expressed factor factor-group finite order given group G group of degree group of finite group of isomorphisms group of linear group of order group whose order Hence identical operation imprimitive systems independent infinite number integer invariant inverse irreducible components irreducible group irreducible representation linear substitutions Moreover multiple necessarily number of distinct number of operations number of symbols operation of G operation of order order of G order pm rational functions relations representation of G represented root of unity self-conjugate operations self-conjugate sub-group set of conjugate set of operations shew shewn simple group simply isomorphic sub-group H sub-group of G sub-group of order Suppose symbol unchanged symmetric group THEOREM tions transformed transitive permutation-group transitive sets variables white polygon zero