The influence of some primary subgroups on the structure of finite groups

Suppose that G is a finite group and H is a subgroup of G. H is said to be an ss-quasinormal subgroup of G if there is a subgroup B of G such that G= HB and H permutes with every Sylow subgroup of B; H is said to be c-normal in G if G has a normal subgroup T such that G= HT and [InlineEquation not available: see fulltext.], where HG is the normal core of H in G. We fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying 1 < | D| < | P| and study the structure of G under the assumption that every subgroup H of P with | H| = | D| is either ss-quasinormal or c-normal in G. Some recent results are generalized and unified. © 2017, Springer International Publishing AG.