The Theory of Groups: An Introduction

3-cycles a₁ automorphism B₁ 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 elements of G endomorphism equivalence extension factor set field F finite group follows free abelian group free group free product function functor G and H G contains G₁ group G group of order H₁ H₂ hence Hint homomorphism ideal induction infinite integers irreducible isomorphism theorem K₁ kernel Lemma Let F Let G Let H linear matrix multiplicative group nilpotent nonabelian group nonempty nonzero element normal subgroup notation number of elements p-group p-primary p-sylow subgroup permutation phism polynomial prime Proof Let Prove that G quotient reader relations semidirect product semigroup simple groups solvable group subgroup H subgroup of G subnormal series subset Suppose Sylow transvection unique vector space word problem