Finite Groups with Given Systems of m-S-Complemented Subgroups

Abstract

Let G be a finite group and H a subgroup of G. We say that H: is generalized S-quasinormal in G if H=leftlangle A,Brightrangle for some modular subgroup A and S-quasinormal subgroup B of G; m-S-complemented in G if there are a generalized S-quasinormal subgroup S and a subgroup T of G such that G=HT and Hcap Tle Sle H. In this paper, we study finite groups with given systems of m-S-complemented subgroups. In particular, we prove that if {mathfrak {F}} is a saturated formation containing all supersoluble groups and E is a normal subgroup of a finite group G such that G/Ein {mathfrak {F}} and for every non-cyclic Sylow subgroup P of E every maximal subgroup of P not having a nilpotent supplement in G is m-S-complemented in G, then Gin {mathfrak {F}} .