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

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.
