Bulletin

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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

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