The Influence Of The Properties Of Maximal Subgroups On The Structure Of A Finite Group

Let G be a finite group and H a subgroup of G. His called s-seminor-mal in G if it permuts with every Sylow p- subgroup of G with (p,|H|) = 1; a subgroup H of G is said to be c - supplemented in G if there exists a subgroup N of G such that G = HN andH(?)N≤H_G= Core_G(H) .In this paper we prove the following result:Theorem Let F be a saturated formation containing U, the class of all supersol-vable groups and G a group with a normal subgroup H such that G/H∈F . Then G∈F if one of following holds: (1) every maximal subgroup of any Sylow subgroup of H is either s - seminormal or c- supplemented in G; (2) every maximal subgroup of any Sylow subgroups of F~*(H), the generalized Fitting subgroup of H, is either s-seminormalor c- supplemented in G. This unifies some recent results.