An extension of the Glauberman ZJ-Theorem

Abstract: 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 $Z(J_o(D))$, $\Omega(Z(J_r(D)))$ and $\Omega(Z(J_e(D)))$ 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 $Z(J_o(D))$, $\Omega(Z(J_r(D)))$ and $\Omega(Z(J_e(D)))$ 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.