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2 Nilpotent Groups
Fundamental Theorems 86
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7r-group A-invariant abelian group abelian normal subgroup abelian subgroup assumption automorphism central product central series CG(A CG(t CG(u CG(x Chapter characteristic subgroup Clearly commutator subgroup conjugacy classes conjugate contradicts coset cyclic group defined definition denote derived group direct product element g element of G element of order elementary abelian extraspecial factor group finite group following propositions Frobenius functor G contains G satisfies group G group of order H of G Hence Hint holds homomorphism implies inductive hypothesis integer involution irreducible characters irreducible representation Let G Let H matrix maximal subgroup minimal normal subgroup NG(P NG(Q nilpotent group nonabelian noncyclic nonidentity element nontrivial odd order Op(G p-group p-nilpotent p-solvable p-stable proof of Theorem proper subgroup Prove the following quaternion group satisfies condition shows simple group solvable group Sp-subgroup of G subgroup H subgroup of G subgroup of order subset subspace Suppose Sylow Sylow subgroup ZJ(P