An extension of the Glauberman ZJ-Theorem

Muhammet Yasir Kızmaz

paper

An extension of the Glauberman ZJ-Theorem

arXiv:1903.08286

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 $P \in S y l_{p} (G)$ . 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 $Qd (p)$ -free group and $P \in S y l_{p} (G)$ . 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.