the following questions come from tiling problems. Maybe the answer is easy but so far I don't know how to start as I can't see whether I have to prove that it is true or to find a counter-example.
Consider a finitely generated group $(G,*)$ and a finite partition of it $(E_0,E_1,...,E_n)$, where $E_0$ contains the neutral of $G$. Assume that for any $i$, there exists $g\in G$ such that $E_i=g*E_0$. Is it true that $E_0$ is a subgroup of $G$? If the answer is no, is it true with additional properties on $G$ (abelian, finite, free, torsion-free...) ?
Thanks by advance for any comment!
It's not the case for arbitrary groups, and abelian and finite doesn't help. For instance, pick a nonzero element $a$ in $\mathbb Z_2^n$ and form pairs of elements that differ by $a$; then pick one from each pair. The resulting set containing the identity will generally not be a group; e.g. in $\mathbb Z_2^3$ with $a=(1,0,0)$ the set $\{(0,0,0),(0,1,0),(0,0,1),(1,1,1)\}$ isn't a group.