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On the Structure of the Mislin Genus of a Pullback

Author

Listed:
  • Thandile Tonisi

    (Department of Mathematics, The University of the Witwatersrand, Johannesburg 2001, South Africa)

  • Rugare Kwashira

    (Department of Mathematics, The University of the Witwatersrand, Johannesburg 2001, South Africa)

  • Jules C. Mba

    (School of Economics, University of Johannesburg, Johannesburg 2006, South Africa)

Abstract

The notion of genus for finitely generated nilpotent groups was introduced by Mislin. Two finitely generated nilpotent groups Q and R belong to the same genus set G ( Q ) if and only if the two groups are nonisomorphic, but for each prime p , their p-localizations Q p and R p are isomorphic. Mislin and Hilton introduced the structure of a finite abelian group on the genus if the group Q has a finite commutator subgroup. In this study, we consider the class of finitely generated infinite nilpotent groups with a finite commutator subgroup. We construct a pullback H t from the l -equivalences H i → H and H j → H , t ≡ ( i + j ) m o d s , where s = | G ( H ) | , and compare its genus to that of H . Furthermore, we consider a pullback L of a direct product G × K of groups in this class. Here, we prove results on the group L and prove that its genus is nontrivial.

Suggested Citation

  • Thandile Tonisi & Rugare Kwashira & Jules C. Mba, 2023. "On the Structure of the Mislin Genus of a Pullback," Mathematics, MDPI, vol. 11(12), pages 1-11, June.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:12:p:2672-:d:1169399
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