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.

中文翻译：

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格劳伯曼 ZJ 定理的扩展

让p是一个奇数素数并让Ĵ○(X),Ĵr(X)和Ĵe(X)表示三个不同版本的 Thompson 子群p-团体X. 在本文中，我们首先证明了格劳伯曼替换定理 [G. 格劳伯曼，a 的特征子群p-稳定组，加拿大。J.数学。 20(1968) 1101–1135，定理 4.1]。其次，我们证明如下： 让G做一个p-稳定组和磷 ∈赛尔p(G). 假设CG(○p(G)) ≤ ○p(G). 如果D是一个强闭子群磷， 然后Z(Ĵ○(D)),Ω(Z(Ĵr(D)))和Ω(Z(Ĵe(D)))是正规子群G. 第三，我们展示以下内容：让G做一个Qd(p)- 自由组和磷 ∈赛尔p(G). 如果D是一个强闭子群磷，然后是子组的归一化器Z(Ĵ○(D)),Ω(Z(Ĵr(D)))和Ω(Z(Ĵe(D)))控制力强G-融合在磷. 我们也证明了一个类似的结果p-稳定且p-约束组。最后，我们给出一个p-nilpotency 标准，它是 Glauberman-Thompson 的扩展p- 幂等性定理。