Suppose that the Frobenius morphism of is an isomorphism. Then it generates a subgroup of the automorphism group of . If is the spectrum of a finite field, then its automorphism group is the Galois group of the field over the prime field, and the Frobenius morphism and its inverse are both generators of the automorphism group. In addition, and may be identified with . The arithmetic and geometric Frobenius morphisms are then endomorphisms of , and so they lead to an action of the Galois group of "k" on "X".