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Keywords:
finite group; $m$-cyclic group; cyclic subgroup
finite group; $m$-cyclic group; cyclic subgroup
Summary:
A finite group $G$ is called an $m$-cyclic group if it has exactly $m$ cyclic subgroups (including the identity subgroup). For $2\leq m\leq 12$, the $m$-cyclic groups have been classified in a series of papers. We push the above research work further to classify the finite \hbox {13-cyclic} groups, which could be considered as a step to answer the open problem posed by M. Tărnăuceanu (2015). The detailed structure of many groups of ``small'' orders is also analyzed. The following main theorem is proved: Let $G$ be a finite 13-cyclic group. Then $|\pi (G)|\leq 2$, and one of the following holds: \begin {itemize} \item [(1)] $|\pi (G)|=1$, $G\cong Q_{32},{\bf Z}_{11}\times {\bf Z}_{11}$ or ${\bf Z}_{p^{12}}$ with $p$ a prime. \item [(2)] $|\pi (G)|=2$, $G\cong D_{22}$, ${\rm SL}(2, 3)$, ${\bf Z}_{11}: {\bf Z}_{5}$, ${\bf Z}_7 : {\bf Z}_8$, ${\bf Z}_7 : {\bf Z}_{27}$, ${\bf Z}_5 : {\bf Z}_{16}$, or ${\bf Z}_3 : {\bf Z}_{32}$. \end {itemize}
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