On the CSS-subgroups and the supersolvabilty of a finite group

Recall that a subgroup H of a finite group G is said to be a CSS-subgroup of G if there is a normal subgroup T of G such that \(G=HT\), and \(H\cap T\) is SS-quasinormal in G. Clearly, both the c-normal subgroups and the SS-quasinormal subgroups are the CSS-subgroups, but the converse does not hold. In this paper, based on the research of c-normality and SS-quainormality, with the help of the group formation theory, we investigate the supersolvabilty of G by assuming that the subgroups with order |D| are CSS-subgroups of G, where |D| is a prime power. Some relevant results to the c-normality and the SS-quainormality are generalized.