Transitive affine transformations on groups

An affine transformation T on a group G is an automorphism followed by a translation; T is transitive if for each JC, y E. G there is an integer n such that T n (x) = y. All groups with transitive affine transformations are determined: the infinite cyclic and infinite dihedral group are the only infinite examples; while the finite examples are semi-direct products of certain odd-order groups by a cyclic, dihedral or quaternion 2-group. The automorphism groups of the above groups are described, and the automorphisms which occur as parts of transitive affine transformations are given. COROLLARY 2.2. Let G be a noncyclic group of order 4. Then an automorphism of G is the associated automorphism of a transitive