关于有限群的$CAP$-嵌入子群

On $CAP$-Embedded Subgroups in Finite Groups

中文关键词:  $CAP$-嵌入子群  极大子群  $p$-幂零群  超可解群.

英文关键词:$CAP$-embedded subgroups  maximal subgroups  $p$-nilpotent groups  supersolvable groups.

中文摘要:

      群$G$的子群$H$称为$G$的$CAP$-嵌入子群, 如果对于$H$的每个素因子$p$, 存在$G$的某个$CAP$-子群$K$使$H$的某个Sylow $p$-子群也是$K$的一个Sylow $p$-子群. 本文通过假定$G$的某些素数幂阶子群是$G$的$CAP$-嵌入子群, 获得一些新的结果.

英文摘要:

      A subgroup $H$ of a finite group $G$ is said to be $CAP$-embedded subgroup of $G$ if, for each prime $p$ dividing the order of $H$, there exists a $CAP$-subgroup $K$ of $G$ such that a Sylow $p$-subgroup of $H$ is also a Sylow $p$-subgroup of $K$. In this paper some new results are obtained based on the assumption that some subgroups of prime power order have the $CAP$-embedded property in the group.