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A group $G$ is called virtually cyclic if it has a cyclic subgroup of finite index. Why are virtually cyclic groups finitely generated?

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Let $K=\left<a\right>$ be your cyclic subgroup of finite index. Now since $K$ is of finite index $n$ in $G$ you have $$G= \bigcup_{i=1}^{n} k_iK $$ Now you can see $G=\left<k_1, k_2, \ldots ,k_n,a\right>$

EDIT Notice that we only used that the subgroup of finite index is finitely generated.

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is $i=1$ there? :-) –  B. S. Sep 11 '12 at 18:16

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