On weakly M-supplemented primary subgroups of finite groups

A subgroup H of a group G is said to be weakly M-supplemented in G if there exists a subgroup B of G provided that (1) G = HB, and (2) if H1/HG is a maximal subgroup of H/HG , then H1B = BH1 < G. where HG is the largest normal subgroup of G contained in H. In this paper we will prove that: Let F be a saturated formation containing all supersolvable groups and G be a group with a normal subgroup H such that G/H ∈ F. If every maximal subgroup of every noncyclic Sylow subgroup of F∗(H) having no supersolvable supplement in G, is weakly M-supplemented in G, then G ∈ F.