# Characterizations of hypercyclically embedded subgroups of finite groups

### Xiaolan Yi

Zhejiang University of Science and Technology, Hangzhou, China

## 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 $a \in (G' \cap F^{*}(E))E'$, $p'$-element $b \in G$ and $q$-element $c \in G'$ we have $[a, b^{p-1}]=1=[a^{q-1}, c]$. Some known results are generalized. \end{abstract}

## Cite this article

Xiaolan Yi, Characterizations of hypercyclically embedded subgroups of finite groups. Rend. Sem. Mat. Univ. Padova 135 (2016), pp. 195–206

DOI 10.4171/RSMUP/135-11