Automorphisms of Z^d-subshifts of finite type

Let (S,s) be a Zd-subshift of finite type. Under a strong irreducibility condition (strong specification), we show that Aut(S) contains any finite group. For Zd-subshifts of finite type without strong specification, examples show that topological mixing is not sufficient to give any finite group in the automorphism group in general: in particular, End(S) may be an abelian semigroup. For an example of a topologically mixing Z2-subshift of finite type, the endomorphism semigroup and automorphism group are computed explicitly. This subshift has periodic-point permutations that do not extend to automorphisms.