New characterizations of a normal subgroup to be hypercyclically embedded

A normal subgroup $E$ of a group $G$ is said to be hypercyclically embedded in $G$ if either $E=1$ or $E\mathrm{\ne }1$ and every chief factor of $G$ below $E$ is cyclic. In this article, we present some new characterizations of a normal subgroup to be hypercyclically embedded. Some recent results in this literature are generalized and unified.