For any nontrivial abelian group $\mathbb{ X}$ we construct a reversible (bireversible in case the order of $\mathbb{ X}$ is odd) automaton such that its set of states and alphabet are identified with $\mathbb{ X}$, transition and output functions are defined via the left and the right regular actions correspondingly and its group splits into the restricted wreath product $\mathbb{ X} \wr \mathbb{ Z}$, i.e. is a lamplighter group.
Categories: math.GR