# A new criterion for finite non-cyclic groups

Research paper by **Wei Zhou, Wujie Shi, Zeyong Duan**

Indexed on: **19 Sep '05**Published on: **19 Sep '05**Published in: **Mathematics - Group Theory**

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#### Abstract

Let $H$ be a subgroup of a group $G$. We say that $H$ satisfies the power
condition with respect to $G$, or $H$ is a power subgroup of $G$, if there
exists a non-negative integer $m$ such that $H=G^{m}=<g^{m} | g \in G >$. In
this note, the following theorem is proved: Let $G$ be a group and $k$ the
number of non-power subgroups of $G$. Then (1) $k=0$ if and only if $G$ is a
cyclic group(theorem of F. Sz$\acute{a}$sz) ;(2) $0 < k <\infty$ if and only if
$G$ is a finite non-cyclic group; (3) $k=\infty$ if and only if $G$ is a
infinte non-cyclic group. Thus we get a new criterion for the finite non-cyclic
groups.