关于有限群的S-半置换子群

On $S$-Semipermutable Subgroups of Finite Groups

中文关键词:  $S$-半置换子群  $p$-幂零群  $p$-超可解群.

英文关键词:$S$-semipermutable subgroups  $p$-nilpotent groups  supersolvable groups.

中文摘要:

      设$d$是有限$p$-群的最小生成元的个数,${\cal M}_d (P)=\{P_{1}, \ldots, P_{d} \}$ 为$P$的满足$\bigcap _{i=1}^{d}P_{i}=\Phi(P)$的极大子群的集合. 在本文，我们研究${\cal M}_{d}(G_{p})$的每个成员的$S$-半置换性对有限群$G$结构的影响,其中$p$是$|G|$的任意素因子，$G_{p}$是$G$的Sylow $p$-子群.

英文摘要:

      Let $d$ be the smallest generator number of a finite $p$-group $P$ and let ${\cal M}_d (P)=\{P_{1}, \ldots, P_{d} \}$ be a set of maximal subgroups of $P$ such that $\bigcap _{i=1}^{d}P_{i}=\Phi(P)$. In this paper, we study the structure of a finite group $G$ under the assumption that every member in ${\cal M}_{d}(G_{p})$ is $S$-semipermutable in $G$ for each prime divisor $p$ of $|G|$ and a Sylow $p$-subgroup $G_{p}$ of $G$.