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.
Isomorphic subgroups, finite index, infinite index
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group-theory
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2Nonabelian free groups give finitely generated examples, as opposed to taking infinite products as suggested in the answers. – 2012-09-10