I'm a hobbyist mathematician so any question I ask here might be at risk of closure. I hope this one is good enough, but I'm not sure. This is a continuation of two questions I asked on math.stackexchange.com: About translating subsets of $\Bbb R^2$ and About translating subsets of $\Bbb Z$. The question is
For which abelian groups $M$ can we find $A,B\subseteq M$ such that
- $A$ is a union of translated (only translations are allowed) copies of $B$;
- $B$ is a union of translated copies of $A$;
- $A$ is not a a single translated copy of $B$ (and the other way around, which follows)?
David Moews showed in an answer to the first question that such sets exist in $\Bbb Z^\omega$ and therefore in any group in which $\Bbb Z^\omega$ embeds. Certainly no such sets can be found in finite groups. Can we describe all groups that have such two sets?