If two groups are isomorphic to subgroups of each other, are they isomorphic themselves? Suppose we have two groups, $G,H$ and that $\rho:G\rightarrow H, \tau:H\rightarrow  G$ are $1-1$ morphisms. Is it true then that $G,H$ are isomorphic? I don't think so, but I can't come up with any counter-example. This definitely holds for finite groups obviously. But what about infinite groups? Moreover, if it doesn't hold in general, does it hold for some large class of groups?

One counterexample: the free group on two letters contains a subgroup isomorphic to the free group on three letters. (see here: Show that the free group on three generators is a subgroup of the free group on two generators)
