On Two Theorems of Thompson

Abstract

Let G be a finite group. THEOREM. Let P E Sylp(G) with Q%(P) < Z(P). If NG(Z(P)) has a normal p-complement, then so does G. COROLLARY. Let M be a nilpotent maximal 8ufgrp of G and P E Syl2 (M) uzth i Q2(P) < Z(P). Then G is solvable. This extends Thompson's solvability theorem [9]. We also give two other results generalizing Thompson's theorem. In this note, we prove some theorems, one of which is similar to Thompson's normal p-complement theorem [8] and the others are generalizations of Thompson's theorem regarding solvability of finite groups [9]. For simplicity we write X E Npi which means the finite group X has a normal p-complement for a prime p. Our other notations are standard and follow [5]. THEOREM 1. Let G be a finite group and P E Sylp(G). If 1(P) < Z(P) and NG(P), CG(Z(P)) E Np, then G E Np. PROOF. Let G be a counterexample of minimal order of the theorem. Then we deduce a contradiction step-by-step. 1. Op,(G) = 1 by the minimality of I10. 2. Op (G) t 1, since otherwise we would have NG(P1) < G for each subgroup of P1 of P and NG(Pl) E Np. In fact, we have a series of subgroups NG (Pi) ), NG (P2 )1, I NG (Pi),I NG (P), where Pi+1 E Sylp(NG(Pi)). Now suppose NG(P1) ? Np but NG(P) E Np. There is some io such that NG(Pi.) ? Np while NG(Pi.+1) E Np. Then we have Q1 (Pi.+i ) < Q1 (P) < Z(P) < Z(Pio+ ) X Pi0+ E Sylp(NG (Pio))i CG(Z(Pio+1)) ? CG(Z(P)) E Np. Thus NG(Pio) E Np by the minimality of IGI. This is a contradiction. So NG(Pl) E Np for each P1 < P and hence G E Np by Frobenius' theorem [5, Theorem 7.45]. This contradicts our hypothesis of G. 3. Set H = Op (G). Then G/H E Np by a discussion similar to step 2. Let K/H be the normal p-complement of G/H. Then 1 <-H < K < G and G is p-solvable. Since Op,(G) = 1, CG(H) < H by Theorem 6.3.2 of [5]. Particularly Z(P) < H. 4. Let M be a maximal subgroup of G containing P. Then M E Np and Op,(M)H = Op(M) x H. This shows Op,(M) < CG (H) < H and so Op(M) = 1. Received by the editors October 25, 1985. Presented at the International Symposium on Group Theory in Beijing in 1984 sponsored by the Ministry of Education of the People's Republic of China. 1980 Mathematics Subject Classification (1985 Revision). Primary 20D20. (?)1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page