Finite $p$-nilpotent groups with some subgroups weakly $\mathcal {M}$-supplemented

Dong, Liushuan

Czechoslovak Mathematical Journal, Tome 70 (2020) no. 1, pp. 291-297.

Résumé

Suppose that $G$ is a finite group and $H$ is a subgroup of $G$. Subgroup $H$ is said to be weakly $\mathcal {M}$-supplemented in $G$ if there exists a subgroup $B$ of $G$ such that (1) $G=HB$, and (2) if $H_{1}/H_{G}$ is a maximal subgroup of $H/H_{G}$, then $H_{1}B=BH_{1}$, where $H_{G}$ is the largest normal subgroup of $G$ contained in $H$. We fix in every noncyclic Sylow subgroup $P$ of $G$ a subgroup $D$ satisfying $1|D||P|$ and study the $p$-nilpotency of $G$ under the assumption that every subgroup $H$ of $P$ with $|H|=|D|$ is weakly $\mathcal {M}$-supplemented in $G$. Some recent results are generalized.

Détail

DOI : 10.21136/CMJ.2019.0273-18

Classification : 20D10, 20D20
Keywords: $p$-nilpotent group; weakly $\mathcal {M}$-supplemented subgroup; finite group