Characterizations of hypercyclically embedded subgroups of finite groups

Abstract

A normal subgroup $H$ of a finite group $G$ is said to be hypercyclically embedded in $G$ if every chief factor of $G$ below $H$ is cyclic. Our main goal here is to give new characterizations of hypercyclically embedded subgroups. In particular, we prove that a normal subgroup $E$ of a finite group $G$ is hypercyclically embedded in $G$ if and only if for every different primes $p$ and $q$ and every $p$-element $a\in (G^{\prime }\cap F^{\ast }(E))E^{\prime }$, $p^{\prime }$-element $b\in G$ and $q$-element $c\in G^{\prime }$ we have $[a,b^{p-1}]=1=[a^{q-1},c]$. Some known results are generalized. \end{abstract}