On $S$-Semipermutable Subgroups of Finite Groups

Abstract:

      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 $G$.