Non-cancellation group of a direct product

The non-cancellation set of a group G, denoted by χ(G), is defined to be the set of all isomorphism classes of groups H such that G×Z ∼= H×Z. While investigating when Z can be cancelled in this direct product, χ(G) has become the focus of many studies. For the semidirect product Gi = Zni ⊥ωi Z, i = 1, 2, methods for computation of the non-cancellation groups χ(G1 × G2), χ(Gki ), k ϵ N and χ(Gi, hi) have been developed. We present in this study, a general method of computing χ(G1 × G2, h), where h : F ,→ G1 ⊆ G1 × G2 and F a finite group.