Conjugacy classes of the automorphism group of a finite group

Abstract

Let K denote the class of semi-direct products of the form R = T oλ Z k , where k ∈ Z +, T is a finite abelian group and λ : Z k → Aut(T) is a group action. The non-cancellation set of R, denoted by χ(R) is the set of all isomorphism classes of groups Q such that Q × Z ∼= R × Z. Various authors have studied the non-cancellation in the class of K-groups. They have described the non-cancellation through conjugacy classes of Aut(T) where T is a finite abelian group. We extend the study of the non-cancellation phenomena in the class of K-group to the groups of the form R = T oλ Z k where T is a finite group but not necessarily abelian. We establish a link between the Nielsen equivalence classes and the conjugacy classes of Aut(T). We then discuss the description of the non-cancellation set χ(R) through Nielsen equivalence classes.