On $\mathcal{M}$-permutable sylow subgroups of finite groups

Abstract

A $p$-subgroup $P\neq 1$ of $G$ is called $\mathcal{M}$-permutable in $G$ if there exists a set $\mathcal{ M}_d(P)=\{P_1,\ldots,P_d\}$ of maximal subgroup $P_i$ of $P$ and a subgroup $B$ of $G$ such that: (1) $\bigcap_{i=1}^{d}{P_i}=\Phi(P)$ and $|P : \Phi(P)|=p^d$; (2) $G=PB$ and $P_iB=BP_i \lt G$ for any $P_i$ of $\mathcal{M}_d(P)$. In this paper, we investigate the influence of $\mathcal M$-permutability of Sylow subgroups in finite groups. Some new results about supersolvable groups and formations are obtained.