LAUSR

Let G be a finite group and H a subgroup of G. We say that H: is generalized S-quasinormal in G if \(H=\left\langle A,B\right\rangle \) 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 \(H\cap T\le S\le H\). 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 \(G/E\in {\mathfrak {F}}\) 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 \( G\in {\mathfrak {F}}\) .