有限群的$\Phi$-$\tau$-可补子群

On $\Phi$-$\tau$-Supplement Subgroups of Finite Groups

中文关键词:  Sylow子群  次正规子群  子群算子  $p$-幂零群  $\Phi$-$\tau$-可补子群

英文关键词:Sylow subgroups  subnormal subgroups  subgroup functor  $p$-nilpotent group  $\Phi$-$\tau$-supplement

中文摘要:

      假设$\tau$是一个子群算子, $H$是有限群$G$的一个$p$-子群. 令 $\bar{G}=G/H_{G}$且$\bar{H}=H/H_{G}$, 如果$\bar{G}$有一个次正规子群$\bar{T}$ 和一个包含于$\bar{H}$ 的$\tau$-子群$\bar{S}$满足$\bar{G}=\bar{H}\bar{T}$且$\bar{H}\cap\bar{T}\leq \bar{S}\Phi(\bar{H})$, 就称$H$是$G$的一个$\Phi$-$\tau$- 可补子群. 文章通过讨论群$G$的准素数子群的$\Phi$-$\tau$-可补性给出了超循环嵌入和$p$-幂零性的一些新的特征.

英文摘要:

      Let $\tau$ be a subgroup functor and $H$ a $p$-subgroup of a finite group $G$. Let $\bar{G}=G/H_{G}$ and $\bar{H}=H/H_{G}$. We say that $H$ is $\Phi$-$\tau$-supplement in $G$ if $\bar{G}$ has a subnormal subgroup $\bar{T}$ and a $\tau$-subgroup $\bar{S}$ contained in $\bar{H}$ such that $\bar{G}=\bar{H}\bar{T}$ and $\bar{H}\cap\bar{T}\leq \bar{S}\Phi(\bar{H})$. In this paper, some new characterizations of hypercyclically embedability and $p$-nilpotency of a finite group are obtained based on the assumption that some primary subgroups are $\Phi$-$\tau$-supplement in $G$.