More on Groups and Counter Automata

Elder, Kambites, and Ostheimer showed that if the word problem of a finitely generated group $H$ is accepted by a $G$-automaton for an abelian group $G$, then $H$ is virtually abelian. We give a new, elementary, and purely combinatorial proof to the theorem. Furthermore, our method extracts an explicit connection between the two groups $G$ and $H$ from the automaton as a group homomorphism from a subgroup of $G$ onto a finite index subgroup of $H$.