/sci/ - Science & Math

I'm having some problems with part (b) of this exercise. It's fairly easy to show that if [math]\Pi_1, \Pi_2, \ldots[/math] are the cosets of a subgroup of [math]\mathbb{Z}[/math], then such a partition is compatible with +, but I don't know what to do next.
I suspect that these are the only compatible partitions, and if so, I was trying to prove this using the result from exercise 8.12 (if S is a subset of G and the cosets of S partition G, then S is a subgroup of G). But I'm not getting anywhere. How do I do solve this problem?