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1-1 function 2-transitive A₁ Abelian group Abelian subgroup Alt(A Alt(n Aut(G automorphism b₁ characteristic commutative 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 is solvable group G group of order H₁ H₂ Hall subgroup Hence G Hom(G infinite Inn(G integers irreducible characters isomorphism Ker(T lemma Let G Let H matrix maximal proper subgroup morphism nilpotent group non-Abelian nontrivial normal non-E subgroup normal subgroup p-group P₁ permutation group primitive S-ring Proof Prove representation of G simple groups solvable group subdivision ring subgroup H subgroup of G subset supersolvable Suppose Sylow p-subgroup Sym(A Sym(M Sym(n T₁ transitive vector space WLOG x₁