부산대학교 도서관

Let $G$ be a finite group, ${\scr A}(G)$ the set of all abelian subgroups of $G$ of maximal order and $J(G)=\langle {\scr A}(G)\rangle$ the Thompson subgroup of $G$. If $H$ is a subgroup of $G$, $C^*_G(H)$ is the generalized centralizer of $H$ in $G$, as defined by Bender. Let ${\scr F}$ be a class of groups. An ${\scr F}$-injector of the group $G$ is a subgroup $H$ of $G$ such that $H\cap N\in{\scr F}$ for each subnormal subgroup $N$ of $G$. The group $G$ satisfies the property $C^N_{\scr F}$ if every subnormal subgroup of $G$ has a unique conjugacy class of ${\scr F}$-injectors. A Fitting class is a class of groups closed under taking normal subgroups and performing normal products. For a Fitting class ${\scr F}$, ${\rm char}({\scr F})=\{p\colon\ p$ prime and there exists $G\in{\scr F}$ such that $p\mid o(G)\}$ and $G_{\scr F}$ is the subgroup of $G$ generated by all the subnormal ${\scr F}$-subgroups of $G$. The main result of the authors is the following generalization of the Glauberman result on $Z(J(P))$, where $P$ is a Sylow $p$-subgroup of $G$. Let ${\scr F}$ be a Fitting class with ${\rm char}({\scr F})=\pi$. Let $G$ be a $p$-stable group where $p\in \pi$ is an odd prime. Suppose that $G$ satisfies the property $C_{\scr F}^N$ and that $C^*_G(G_{\scr F})\leq F(G)$, the Fitting subgroup of $G$. Let $H$ be an ${\scr F}$-injector of $G$ and assume that for every $A\in{\scr A}(H)$ there exists a positive integer $n$ such that $[O_p(G),A,{\overset(n)\to\cdots},A]$ is abelian. Then $O_p(Z(J(H)))$ is a normal subgroup of $G$.