A Characteristic Subgroup of an $E_{\pi}^n$ Group

Abstract

Suppose G is a group which is p-stable and p-constrained, where p is an odd prime, and S is a Sylow p-subgroup of G. A classical theorem of Glauberman shows that ZJ(S)Op′(G)◃G, where J(S) is the Thompson subgroup of S. In this paper, we generalize the above result by replacing the odd prime p with a set π of odd primes. More precisely, suppose π is a set of odd primes and G is an [Formula: see text] group which is π-stable and π-constrained. We prove that if H∈ Hallπ(G) then ZJ(H)Oπ′(G)◃G and G=NG(ZJ(H))Oπ′(G). An interesting application is also given.