Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question

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.

share|cite|improve this answer
is $i=1$ there? :-) – Babak S. Sep 11 '12 at 18:16

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.