# Rendiconti del Seminario Matematico della Università di Padova

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**Volume 135, 2016, pp. 195–206**

**DOI: 10.4171/RSMUP/135-11**

Published online: 2016-05-19

Characterizations of hypercyclically embedded subgroups of finite groups

Xiaolan Yi^{[1]}(1) Zhejiang University of Science and Technology, Hangzhou, China

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}

*Keywords: *Finite group, supersoluble group, hypercyclically embedded subgroup, Sylow subgroup, generalized Fitting subgroup

Yi Xiaolan: Characterizations of hypercyclically embedded subgroups of finite groups. *Rend. Sem. Mat. Univ. Padova* 135 (2016), 195-206. doi: 10.4171/RSMUP/135-11