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Transformations and Subgroups
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1-1 function 2-transitive Abelian group Abelian subgroup Aut(G automorphism commutative complement composition series conjugate classes contradiction cosets cyclic groups defined direct sum division ring elementary Abelian endomorphism equivalent example Exercise factor group factor system finite group finite number follows from Theorem Fr(G free Abelian free Abelian group Frobenius group G containing g e G G is solvable group G group of order Hall subgroup Hence G Hom(G infinite Inn(G integers irreducible characters isomorphism Ker(r lemma Let G Let H matrix maximal proper subgroup morphism nilpotent group non-Abelian nontrivial normal subgroup P-subgroup permutation group primitive S-ring Proof Prove representation of G S-group simple groups solvable group subdivision ring subgroup H subgroup of G subset supersolvable Suppose Sylow subgroups Sylp(G Sym(A Sym(M transitive vector space WLOG x e G