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

摘要

If M is a finitely generated group having a finite commutator subgroup, then the set χ(M) of all isomorphism classes of groups G such that G×ℤ≃M×ℤ is a finite set and coincides with the Mislin genus (M) of M if M is nilpotent. For such groups M, there is a group structure on χ(M) defined in terms of the indices of embeddings of G into M, for groups G representing elements of χ(M). Such embeddings do exist and their indices are necessarily finite. If M is nilpotent, then this group structure on χ(M) coincides with the Hilton-Mislin group structure on the genus of M. In this paper we calculate the group χ(H k ) where H k is the direct product of k copies of a group the form H=〈 a,b | a n=1, bab -1=a u 〉, for any relatively prime pair of natural numbers n,u. In particular we find that for each such group H we have an isomorphism χ(H 2)≃χ(H k ) whenever k>2.