The non-cancellation group of a direct power of a (finite cyclic)-by-cyclic group. (Q1882576)

If \(M\) is a finitely generated group having finite commutator subgroup, then the set \(\chi(M)\) of all isomorphism classes of groups \(G\) such that \(G\times\mathbb{Z}\simeq M\times\mathbb{Z}\) is a finite set. For such groups \(M\) there is a group structure on \(\chi(M)\). In this article the author calculates \(\chi(H^k)\) where \(H^k\) is the direct product of \(k\) copies of the group \(H=\langle a,b\mid a^n=1,\;bab^{-1}=a^u\rangle\). In particular, the following isomorphism \(\chi(H^k)\simeq\chi(H^2)\) is valid whenever \(k\geq 2\).