Showing the uniqueness of the group of integers

[playa007]

Homework Statement

Show that the infinite cyclic group Z is the unique group that is isomorphic to all its non-trivial proper subgroups

Homework Equations

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?

 

[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?

 

[Hurkyl]

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