On ℳ-supplemented subgroups

Abstract

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