| Abstrakt: |
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. [ABSTRACT FROM AUTHOR] |
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