On the Structure of the Mislin Genus of a Pullback

Thandile Tonisi (), Rugare Kwashira and Jules C. Mba
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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

Mathematics, 2023, vol. 11, issue 12, 1-11

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.

Keywords: mislin genus; noncancellation; short exact sequence; pullback diagram; localization (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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