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 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 M-supplemented.In particular, we show that if k ≥ 2, then G/O p ′ (G) is supersolvable with a normal Sylow p-subgroup and a cyclic p-complement.