Isomorphic subgroups, finite index, infinite index Is it possible to have a group $G,$ which has two different, but isomorphic subgroups $H$ and $H',$ such that one is of finite index, and the other one is of infinite index?
If not, why is that not possible. If there is a counterexample please give one.

Yes. For $n\in\Bbb N$ let $G_n$ be a copy of $\Bbb Z/2\Bbb Z$, and let $G=\prod_{n\in\Bbb N}G_n$ be the direct product. Then $H_0=\{0\}\times\prod_{n>0}G_n$ is isomorphic to $H_1=\prod_{n\in\Bbb N}A_n$, where $A_n=\{0\}$ if $n$ is odd and $A_n=G_n$ if $n$ is even. Clearly $[G:H_0]=2$ and $[G:H_1]$ is infinite.
Of course $H_0$ and $H_1$ are both isomorphic to $G$, so I could have used $G$ instead of $H_0$, but I thought that you might like the subgroups to be proper.
