Glauberman ZJ定理的一个推广
An extension of the Glauberman ZJ-Theorem
摘要：
设$p$是奇素数，$J_o(X)$，$J_r(X)$和$J_e(X)$表示$p$ -群$X$的三种不同形式的Thompson子群.本文首先证明了Glauberman替换定理的一个推广。其次，证明了：设$G$是$p$ -稳定群，且$P\in Syl_p(G)$ .假设$C_G(O_{p}(G))\leq O_{p}(G)$ .若$D$是$P$中的强闭子群，则$Z(J_o(D))$、$Ω(Z(J_r(D)))$和$Ω(Z(J_e(D)))$是$G$的正规子群。第三，我们证明了：设$G$是$\text{Qd}(p)$ -自由群，且$P\in Syl_p(G)$ .如果$D$是$P$中的强闭子群，则子群$Z(J_o(D))$、$Ω(Z(J_r(D)))$和$Ω(Z(J_e(D)))$的正规化子控制$P$中的强$G$融合。对于$p$ -稳定和$p$ -约束群，我们也证明了类似的结果.最后，我们给出了一个$p$ -幂零性准则，它是Glauberman-Thompson $p$ -幂零性定理的推广.
英文摘要：
Let $p$ be an odd prime and let $J_o(X)$, $J_r(X)$ and $J_e(X)$ denote the three different versions of Thompson subgroups for a $p$-group $X$. In this article, we first prove an extension of Glauberman's replacement theorem. Secondly, we prove the following: Let $G$ be a $p$-stable group and $P\in Syl_p(G)$. Suppose that $C_G(O_{p}(G))\leq O_{p}(G)$. If $D$ is a strongly closed subgroup in $P$, then $Z(J_o(D))$, $Ω(Z(J_r(D)))$ and $Ω(Z(J_e(D)))$ are normal subgroups of $G$. Thirdly, we show the following: Let $G$ be a $\text{Qd}(p)$-free group and $P\in Syl_p(G)$. If $D$ is a strongly closed subgroup in $P$, then the normalizers of the subgroups $Z(J_o(D))$, $Ω(Z(J_r(D)))$ and $Ω(Z(J_e(D)))$ control strong $G$-fusion in $P$. We also prove a similar result for a $p$-stable and $p$-constrained group. Lastly, we give a $p$-nilpotency criteria, which is an extension of Glauberman-Thompson $p$-nilpotency theorem.
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