具有若干$c_p$-可补子群的有限群

Finite Groups with Some $c_p$-Supplemented Subgroups

中文关键词:  $c$-可补子群  $c_p$-正规子群；$c_p$-可补子群  $CS_p$-群  $p$-超可解群

英文关键词:$c$-supplemented subgroup  $c_p$-normal subgroup  $c_p$-supplemented subgroup  $CS_p$-group  $p$-supersolvable group

中文摘要:

      设$H$为有限群$G$的子群且$p$为整除群$G$的阶的素因子. 我们称$H$在$G$中为$c_p$-可补的,如果$G$中存在$H$的包含$H_G$的补子群$T$使得$H\cap T/H_G$为$p'$-群, 其中$H_G$为$H$在$G$中的核. 群$G$ 称$CS_p$-群, 如果$G$的所有$p$-子群都在$G$中$c_p$-可补. 本文，我们刻画具有若干$c_p$-可补$p$-子群的有限群的$p$-可解性和$p$-超可解性. 此外,我们给出$p$-可解群为$CS_p$-群的若干等价条件. 最后, 我们给出两个$CS_p$-群的直积为$CS_p$-群的判别准则. 我们的结果推广了近期的若干结论.

英文摘要:

      Let $H$ be a subgroup of a finite group $G$ and $p$ a prime divisor dividing the order of $G$. We say $H$ is $c_p$-supplemented in $G$ if there exists a supplement $T$ to $H$ in $G$ containing $H_G$ such that $H\cap T/H_G$ is a $p'$-group, where $H_G$ is the core of $H$ in $G$. A $CS_p$-group is a group in which every $p$-subgroup is $c_p$-supplemented. In this paper, we characterize the $p$-solvability and $p$-supersolvability of groups $G$ with some certain $p$-subgroups being $c_p$-supplemented. Furthermore, we give some equivalent conditions of $CS_p$-group in $p$-solvable universe. Finally, we give some criteria of $CS_p$-groups for the direct product of two $CS_p$-groups. Our results extend some recent conclusions.