# It is true that every group that has a finite number of subgroups is finite? [duplicate]

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It is true that every group that has a finite number of subgroups is finite?

I think not, but I can not find counterexamples.

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## marked as duplicate by Martin BrandenburgJun 21 '14 at 13:31

–  Jérémy Blanc Jun 21 '14 at 13:21

Yes, true - look at $\langle g\rangle$ for every $g \in G$ and note that an infinite cyclic group has an infinite number of subgroups.
$\langle g \rangle$ must be finite, and since $G$ has only a finite number of subgroups, we get that $G=\bigcup_{g \in G}\langle g \rangle$ is a finite union. hence $G$ must be finite after all!
This part of that an infinite cyclic group has an infinite number of subgroups is because each $g^n$ would create a new subgroup $<g^n>$? –  Croos Jun 21 '14 at 13:53