On $\mathcal{M}$-supplemented subgroups

Abstract

Let $G$ be a finite group and $p^k$ be a prime power dividing $|G|$. A subgroup $H$ of $G$ is called to be $\mathcal{M}$-supplemented in $G$ if there exists a subgroup $K$ of $G$ such that $G=HK$ and $H_{i}K<G$ for every maximal subgroup $H_i$ of $H$. In this paper, we complete the classification of the finite groups $G$ in which all subgroups of order $p^k$ are $\mathcal{M}$-supplemented. In particular, we show that if $k\ge 2$, then $G/{\mathrm{O}}_{p^{\prime }}(G)$ is supersolvable with a normal Sylow $p$-subgroup and a cyclic $p$-complement.