有限群的素数幂阶S-拟正规嵌入子群

设G为有限p-可解群,其中p为|G|的奇素因子.若P为G的Sylow p-子群且最小生成系含d个元素.考虑集合M_d(P)={P_1,…,P_d},其中P_1,…,P_d是P的极大子群且满足(?)P_i=φ(P).证明了若M_d(P)中每个元在G中是S-拟正规嵌入的,则G为p-超可解群.作为应用,还得到了一些进一步的结论.

On S-Quasinormally Embedded Subgroups of Prime Power Order in Finite Groups

$p$-solvable finite group, where $p$ is an odd prime divisor of $|G|$, and $P$ be a Sylow $p$-subgroup of $G$ with the smallest generator number $d$. Consider the set ${\cal M}_d(P)=\{P_1,\cdots,P_d\}$, where $P_1,\cdots,P_d$ are the maximal subgroups of $P$ such that $\bigcap\limits_{i=1}^d P_i=\Phi (P)$. It is shown that if every member of ${\cal M}_d(P)$ is $S$-quasinormally embedded in $G$, then $G$ is $p$-supersolvable. As its applications, some further results are obtained.