Images of commutator MAPS

The commutator map θα: G → G associated with the element α of the group G is defined by θα(g) = g −1α−1gα for all g ε G. In this paper we prove that if the image θα is a subgroup of the finite group G, then θα(G) is soluble. (This in fact generalises the well-known result which states that all finite groups that admit a fixed-point-free automorphism are soluble.) An application (that motivated this paper) to the theory of permutation groups is also given.