Volume 36 • Number 4 • 2013   • Finite Groups with Some $\mathcal M$-Permutable Primary Subgroups Hongwei Bao and Long Miao Abstract. Let $d$ be the smallest generator number of a finite $p$-group $P$ and $\mathcal M$$_d(P)=\{P_1,P_2,\cdots P_d\}$ be the set of maximal subgroups of $P$ such that $\bigcap_{i=1}^{d}P_i=\Phi(P)$. Then $P$ is called $\mathcal M$-permutable in a finite group $G$, if there exists a subgroup $B$ of $G$ such that $G=PB$ and $P_iB G $ for every $P_i$ of ${\mathcal M}_d(P)$.In this paper, we investigate the structure of finite groups by some $\mathcal M$-permutable subgroups of the Sylow $p$-subgroup. Some new results about $p$-supersolvable groups and $p$-nilpotent groups are obtained. 2010 Mathematics Subject Classification: Primary: 20D10,20D20 Full text: PDF