. 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, whereJ(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 Enπ 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.