# Showing the uniqueness of the group of integers

1. Sep 9, 2009

### playa007

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?

2. Sep 10, 2009

### lanedance

hi playa007, to be honest I've only had limited exposure to groups, so not sure... but as an idea could you assume you have two different infinite cyclic groups isomorphic to their respective non-trivial proper subgroups, then construct an isomorphism to give a contradiction and show they are in fact the same (unique) group?

3. Sep 10, 2009

### Hurkyl

Staff Emeritus
Well, you could start by trying some specific examples, to get some ideas.