SOME NEW CHARACTERISATIONS OF FINITE -SUPERSOLUBLE GROUPS

Let $G$ be a finite group. A subgroup $H$ of $G$ is said to be $E$ -supplemented in $G$ if there is a subgroup $T$ of $G$ such that $G= HT$ and $H\cap T\leq {H}_{eG} $ , where ${H}_{eG} $ denotes the subgroup of $H$ generated by all those subgroups of $H$ which are $S$ -quasinormally embedded in $G$ . In this paper, some new characterisations of $p$ -supersolubility of finite groups are given under the assumption that some primary subgroups are $E$ -supplemented.