An extension of the glauberman ZJ-theorem

Abstract

Let p be an odd prime and let Jo(X), Jr(X) and Je(X) denote the three different versions of Thompson subgroups for a p-group X. In this paper, we first prove an extension of Glauberman’s replacement theorem [G. Glauberman, A characteristic subgroup of a p-stable group, Canad. J. Math. 20 (1968) 1101–1135, Theorem 4.1]. Second, we prove the following: Let G be a p-stable group and P∈Sylp(G). Suppose that CG(Op(G))≤Op(G). If D is a strongly closed subgroup in P, then Z(Jo(D)), Ω(Z(Jr(D))) and Ω(Z(Je(D))) are normal subgroups of G. Third, we show the following: Let G be a Qd(p)-free group and P∈Sylp(G). If D is a strongly closed subgroup in P, then the normalizers of the subgroups Z(Jo(D)), Ω(Z(Jr(D))) and Ω(Z(Je(D))) control strong G-fusion in P. We also prove a similar result for a p-stable and p-constrained group. Finally, we give a p-nilpotency criteria, which is an extension of Glauberman–Thompson p-nilpotency theorem.