On $\mathcal{M}$-supplemented subgroups

TL;DR

This paper classifies finite groups where all subgroups of a given prime power order are $ ext{M}$-supplemented, revealing structural properties like supersolvability and normal Sylow subgroups for certain cases.

Contribution

It completes the classification of finite groups with all subgroups of order $p^k$ being $ ext{M}$-supplemented, especially for $k eq 1$, and describes their structural characteristics.

Findings

All such subgroups are $ ext{M}$-supplemented in the classified groups.

For $k eq 1$, the quotient $G/\mathrm{O}_{p'}(G)$ is supersolvable.

These groups have a normal Sylow $p$-subgroup and a cyclic $p$-complement.

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 $H_iK<G$ 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 $k\geq 2$, then $G/\mathrm{O}_{p'}(G)$ is supersolvable with a normal Sylow $p$-subgroup and a cyclic $p$-complement.