This paper assumes that an automa A= (S, M, I, f) is composed of the finite nonempty set S, a non-empty set I, a function M on SxI into S, and an operation f defined over I so that the system (I, f) is a semigroup. It is also assumed that M (s, f (x, y)) = M (M (s, x), y) for all s = S and x, y = I, and that A is strongly connected. A function g:A→A is operation preserving if g(M(si, x)) = M (g (si), x) for all si = S and x = I. The set G(A) of all such functions for a given A is a group of regular permutations. AS a result, the order of G(A) divides the order of S. Moreover, for every G of regular permutations of n letters there is an automata A such that G=G(A).There is an interesting relationship between I and G (A).Define the classes Tij of elements of I by x = Tij if and only if (iff) M(si, x)=sj and the operation δ i between these classes by Tij δ i Tik = Tim iff Tij TikTim. Then if G(A) is any subgroup of G(A), there is A set Ti* of classes such that (Ti*, δi) is a group isomorphic to G*(A).An abelian automata A has been defined by Fleck [7]. An n-state automata A is abelian iff G(A) is abelian and of order n. If A is abelian then (I, f) is homomorphic to G(A). If we say that the semigroup (I, f) is acceptable when there is some automata A= (S, M, I, f) then acceptable semigroups have great similarity to groups. Now not all semigroups are acceptabl. A partial characterization of acceptable semigroups is:a11 free semigroups are acceptable, in fact, to abelian automata of arbitrary order.