$G$ is a cyclic group iff 
$$ \forall H < G, \ \exists k, \ H = \{a^k : a \in G\}. $$


Is it right?



A proposition on cyclic group
A bit of searching revealed the following reference for this statement in question



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> [F. Szasz, On cyclic groups](http://matwbn.icm.edu.pl/ksiazki/fm/fm43/fm43123.pdf), Fund. Math., 43(1956), 238-240
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In the following more recent paper the authors proved a refinement. Let $k$ denote the number of subgroups of $G$ that are not of the form $\langle a^n,a\in G\rangle$. The main theorem says that $k=0$ iff $G$ is cyclic, $1\le k<\infty$ iff $G$ is finite non-cyclic, and $k=\infty$ iff $G$ is infinite non-cyclic.



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> [W. Zhoua, W. Shib, Z. Duan, A new criterion for finite non-cyclic groups](http://arxiv.org/abs/math/0509421) Communications in Algebra, 34 (2006), 4453-4457
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