On $$\mathcal {SSH}$$ SSH -subgroups of finite groups

Abstract

Let G be a finite group. A subgroup H of G is s-permutable in G if H permutes with every Sylow subgroup of G. A subgroup H of G is called an $$\mathcal {SSH}$$ -subgroup in G if G has an s-permutable subgroup K such that $$H^{sG} = HK$$ and $$H^g \cap N_K (H) \leqslant H$$ , for all $$g \in G$$ , where $$H^{sG}$$ is the intersection of all s-permutable subgroups of G containing H. We study the structure of finite groups under the assumption that the maximal or the minimal subgroups of Sylow subgroups of some normal subgroups of G are $$\mathcal {SSH}$$ -subgroups in G. Several recent results from the literature are improved and generalized.