The structure of finite groups with weakly c-normal subgroups

For a subgroup of a finite group we introduce a new property called weakly c-normal. Suppose that G is a finite group and H is a subgroup of G. H is said to be weakly c-normal in G if there exists a subnormal subgroup K of G such that (G=HK) and (Hcap K) is s-quasinormally embedded 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 weakly c-normal in G. Some recent results are generalized and unified.