## The Theory of Groups: An Introduction |

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### Contents

GROUPS AND HOMOMORPHISMS | 11 |

Normal Subgroups and Quotient Groups | 21 |

The Correspondence Theorem | 27 |

Copyright | |

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3-cycles automorphism binary operation composition series conjugacy classes conjugate COROLLARY cosets cyclic groups defined Definition Let denote diagram direct product direct sum direct summand disjoint divisible group divisor elements of G endomorphism equivalence exact sequence extension factor set field F finite group follows free abelian group free group free product function functor G and H G contains group G group of order hence homomorphism ideal induction infinite integers irreducible isomorphism theorem kernel Lemma Let G Let H linear matrix module multiplicative group nilpotent nonabelian group nonempty nonzero element normal subgroup notation number of elements one-to-one correspondence p-group p-primary p-sylow subgroup permutation phism polynomial prime Proof Let Prove that G pure subgroup quotient reader relations semidirect product semigroup simple groups solvable group subgroup H subgroup of G subnormal series subset sum of cyclic Suppose Sylow sylow subgroup transvection Turing machine unique vector space word problem