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THE ISOMORPHISM THEOREMS
THE SYLOW THEOREMS
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affine Assume Aut(K automorphism commute composition series conjugacy classes conjugate Corollary cosets cyclic groups defined Definition Let denoted diagram direct product direct sum disjoint divisor equation equivalence exact sequence example Exercise exists factor groups factor set finite group finitely presented group follows free abelian free abelian group free group function G acts G contains G-set group G group of order hence Hint HNN extension homomorphism imbedded implies induction infinite integer kernel Lemma Let G Let H linear matrix multiplicative group nilpotent nonsingular normal series normal subgroup notation number of elements one-one correspondence p-group p-primary permutation polynomial positive words prime Proof Let Prove that G quotient reader recursive relations semidirect product semigroup shows simple group solvable group stable letter Steiner system subgroup H subgroup of G subgroup of order subset subword summand Sylow p-subgroup torsion-free transitive G-set transvection unique vector space