Finite Groups with Some Maximal Subgroups of Sylow Subgroups M-Supplemented

摘要

A subgroup H of a group G is said to be M-supplemented in G if there exists a subgroup B of G such that G = HB and TB < G for every maximal subgroup T of H. In this paper, we obtain the following statement: Let F be a saturated formation containing all supersolvable groups and H be a normal subgroup of G such that G/H ∈ F. Suppose that every maximal subgroup of a noncyclic Sylow subgroup of F~*(H), having no supersolvable supplement in G, is M-supplemented in G. Then G ∈ F.