“…The language family recognized by permutation automata is known as the group languages, because their syntactic monoid is a group, and it has received some attention in the literature on algebraic automata theory [9]. Recently, Hospodár and Mlynárčik [3] determined the state complexity of operations on these automata, while Rauch and Holzer [12] investigated the effect of operations on permutation automata on the number of accepting states. Permutation automata are reversible, in the sense that, indeed, knowing the current state and the last read symbol one can always reconstruct the state at the previous step.…”