# Finite index of a subgroup of an infinite group

It's mentioned in herstein that there can be infinite groups whose subgroups have finite index. I cannot think of any examples. Some examples would be useful

• A related interesting phenomenon is that infinite groups can have subgroups of finite order. The group of non-zero real numbers under multiplication incorporates the subgroup $\{-1, 1\}$ which I thought was quite amazing. Dec 25, 2015 at 10:56
• I am surprised that you cannot think of any examples, since the infinite cyclic group, which must be just aboout the best known infinite group, is an example. Dec 25, 2015 at 11:08
• Yes. I was making a foolish mistake Dec 25, 2015 at 11:51
• The trivial subgroup has infinite index in any infinite group...
– YCor
Dec 28, 2015 at 0:45

• Formally the index of a subgroup is equal to the number of cosets. $\mathbb{Z}/\mathbb{2Z}$ will have two cosets, namely: $0+\mathbb{2Z}$ and $1+\mathbb{2Z}$, therefore the index is 2. Dec 25, 2015 at 11:08