ON AUTOMORPHISMS OF FINITE GROUPS

Abstract

We consider orbits of elements of a finite group G with respect to the action on G of a cyclic automorphism group generated by . We obtain sufficient conditions for the existence of an orbit whose length is equal to the order of the automorphism . Namely, such an orbit exists for any automorphism of a semisimple or nilpotent finite group G and for an automorphism of an arbitrary finite group G when the orders of and G are relatively prime. In the general case, the question of the existence of such an orbit for an automorphism of a finite group is answered negatively; a series of counterexamples is constructed. Nevertheless, the order of an automorphism of a finite group G is in all cases bounded by the order of G. Bibliography: 1 item.