1. The problem statement, all variables and given/known data Show that the infinite cyclic group Z is the unique group that is isomorphic to all its non-trivial proper subgroups 2. Relevant equations 3. The attempt at a solution Due to the fact that Z is cyclic and that every subgroup is a cyclic group, every subgroup of Z is a cyclic group, precisely of the form mZ (where m=>2); and the isomorphism between the two groups is not difficult to show. But I'm wondering how to assert uniqueness that Z is indeed the unique group?