On two theorems of Thompson

Abstract

Let G G be a finite group. Theorem. Let P ∈ Syl p ( G ) P \in {\operatorname {Syl} _p}(G) with Ω 1 ( P ) ≤ Z ( P ) {\Omega _1}(P) \leq Z(P) . If N G ( Z ( P ) ) {N_G}(Z(P)) has a normal p p -complement, then so does G G . Corollary. Let M M be a nilpotent maximal subgroup of G G and P ∈ Syl 2 ( M ) P \in {\operatorname {Syl} _2}(M) with Ω 2 ( P ) ≤ Z ( P ) {\Omega _2}(P) \leq Z(P) . Then G G is solvable. This extends Thompson’s solvability theorem [9]. We also give two other results generalizing Thompson’s theorem.