Finite Group Theory

A-invariant abelian action action of G acts additive apply arbitrary assertion assume automorphisms central Chapter characteristic choose commutator complement conclude conjugate consider construct contains contradiction coprime Corollary corresponding course cyclic deduce defined definition denote direct distinct divides easy elementary abelian equality exactly example exists fact factor Finally finite group fixed follows Frobenius given group G Hall hence holds homomorphism hypothesis identity induction integer isomorphic least Lemma Let G matrix maximal subgroup multiplication NG(P nilpotent nonabelian nonidentity elements nontrivial normal p-complement normal subgroup Note observe orbit p-group particular permutation possible prime primitive problem Proof prove Recall result right cosets simple groups solvable stabilizer Step subgroup of G subnormal suffices suppose Sylow p-subgroup Theorem theory transfer transitive trivial unique wanted write yields