The influence of weakly 3-permutable subgroups on the structure of finite groups

Let 3 be a complete set of Sylow subgroups of a finite group G, that is, for each prime p dividing the order of G, 3 contains exactly one and only one Sylow p-subgroup of G. A subgroup H of G is said to be 3-permutable of G if H permutes with every member of 3. A subgroup H of G is said to be a weakly 3-permutable subgroup of G if there exists a subnormal subgroup K of G such that G = HK and H boolean AND K < H-3, where H3 is the subgroup of H generated by all those subgroups of H which are 3 permutable subgroups of G. We investigate the structure of the finite group G under the assumption that every cyclic subgroup of prime order p or of order 4 (if p = 2) is a weakly 3-permutable subgroup of G. Our results extend and generalize several results in the literature.