子群的S-条件置换性对有限群结构的影响

The Influence of $s$-Conditional Permutability of Subgroups on the Structure of Finite Groups

中文关键词:  有限群  S-条件置换子群  饱和群系  超可解群  幂零群.

英文关键词:finite groups  $s$-conditionally permutable groups  saturated formations  supersoluble groups  nilpotent groups.

中文摘要:

      令 $G$ 是个有限群.固定 $|G|$ 的一个素因子 $p$ 和 $G$ 的一个 Sylow $p$-子群 $P$. 我们用 $d$ 表示 $P$ 的最小生成元的个数且 $\mathcal{M}$$_{d}(P)$ 表示$P$的极大子群$P_{1}, P_{2},\ldots, P_{d}$的生成系其满足等式$\bigcap_{i=1}^{d}=\Phi(P)$, $\Phi(P)$ 为 $P$ 的 Frattini 子群.在这篇文章中,我们将研究某个固定的 $\mathcal{M}$$_{d}(P)$ 中的元素的S-条件置换性对有限群结构的影响.一些新的结果被得到,一些知道的结果被推广.

英文摘要:

      Let $G$ be a finite group. Fix a prime divisor $p$ of $|G|$ and a Sylow $p$-subgroup $P$ of $G$, let $d$ be the smallest generator number of $P$ and ${\cal M}_{d}(P)$ denote a family of maximal subgroups $P_{1}, P_{2}, \ldots, P_{d}$ of $P$ satisfying $\bigcap_{i=1}^{d}P_{i}=\Phi(P)$, the Frattini subgroup of $P$. In this paper, we shall investigate the influence of $s$-conditional permutability of the members of some fixed ${\cal M}_{d}(P)$ on the structure of finite groups. Some new results are obtained and some known results are generalized.