On the Structure of the Mislin Genus of a Pullback
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 <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">G</...
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MDPI AG
2023-06-01
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| Online Access: | https://www.mdpi.com/2227-7390/11/12/2672 |
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| author | Thandile Tonisi Rugare Kwashira Jules C. Mba |
| author_facet | Thandile Tonisi Rugare Kwashira Jules C. Mba |
| author_sort | Thandile Tonisi |
| collection | DOAJ |
| description | 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 <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">G</mi><mo>(</mo><mi>Q</mi><mo>)</mo></mrow></semantics></math></inline-formula> if and only if the two groups are nonisomorphic, but for each prime <i>p</i>, their p-localizations <inline-formula><math display="inline"><semantics><msub><mi>Q</mi><mi>p</mi></msub></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msub><mi>R</mi><mi>p</mi></msub></semantics></math></inline-formula> are isomorphic. Mislin and Hilton introduced the structure of a finite abelian group on the genus if the group <i>Q</i> 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 <inline-formula><math display="inline"><semantics><msub><mi>H</mi><mi>t</mi></msub></semantics></math></inline-formula> from the <i>l</i>-equivalences <inline-formula><math display="inline"><semantics><mrow><msub><mi>H</mi><mi>i</mi></msub><mo>→</mo><mi>H</mi></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><msub><mi>H</mi><mi>j</mi></msub><mo>→</mo><mi>H</mi></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><mi>t</mi><mo>≡</mo><mo>(</mo><mi>i</mi><mo>+</mo><mi>j</mi><mo>)</mo><mspace width="0.166667em"></mspace><mi>m</mi><mi>o</mi><mi>d</mi><mspace width="0.166667em"></mspace><mi>s</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math display="inline"><semantics><mrow><mi>s</mi><mo>=</mo><mo>|</mo><mrow><mi mathvariant="script">G</mi><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>|</mo></mrow></semantics></math></inline-formula>, and compare its genus to that of <i>H</i>. Furthermore, we consider a pullback <i>L</i> of a direct product <inline-formula><math display="inline"><semantics><mrow><mi>G</mi><mo>×</mo><mi>K</mi></mrow></semantics></math></inline-formula> of groups in this class. Here, we prove results on the group <i>L</i> and prove that its genus is nontrivial. |
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| language | English |
| last_indexed | 2024-03-11T02:11:09Z |
| publishDate | 2023-06-01 |
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| series | Mathematics |
| spelling | doaj.art-34b57f0c6cf14376a07489c566b2f7a02023-11-18T11:28:11ZengMDPI AGMathematics2227-73902023-06-011112267210.3390/math11122672On the Structure of the Mislin Genus of a PullbackThandile Tonisi0Rugare Kwashira1Jules C. Mba2Department of Mathematics, The University of the Witwatersrand, Johannesburg 2001, South AfricaDepartment of Mathematics, The University of the Witwatersrand, Johannesburg 2001, South AfricaSchool of Economics, University of Johannesburg, Johannesburg 2006, South AfricaThe 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 <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">G</mi><mo>(</mo><mi>Q</mi><mo>)</mo></mrow></semantics></math></inline-formula> if and only if the two groups are nonisomorphic, but for each prime <i>p</i>, their p-localizations <inline-formula><math display="inline"><semantics><msub><mi>Q</mi><mi>p</mi></msub></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msub><mi>R</mi><mi>p</mi></msub></semantics></math></inline-formula> are isomorphic. Mislin and Hilton introduced the structure of a finite abelian group on the genus if the group <i>Q</i> 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 <inline-formula><math display="inline"><semantics><msub><mi>H</mi><mi>t</mi></msub></semantics></math></inline-formula> from the <i>l</i>-equivalences <inline-formula><math display="inline"><semantics><mrow><msub><mi>H</mi><mi>i</mi></msub><mo>→</mo><mi>H</mi></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><msub><mi>H</mi><mi>j</mi></msub><mo>→</mo><mi>H</mi></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><mi>t</mi><mo>≡</mo><mo>(</mo><mi>i</mi><mo>+</mo><mi>j</mi><mo>)</mo><mspace width="0.166667em"></mspace><mi>m</mi><mi>o</mi><mi>d</mi><mspace width="0.166667em"></mspace><mi>s</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math display="inline"><semantics><mrow><mi>s</mi><mo>=</mo><mo>|</mo><mrow><mi mathvariant="script">G</mi><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>|</mo></mrow></semantics></math></inline-formula>, and compare its genus to that of <i>H</i>. Furthermore, we consider a pullback <i>L</i> of a direct product <inline-formula><math display="inline"><semantics><mrow><mi>G</mi><mo>×</mo><mi>K</mi></mrow></semantics></math></inline-formula> of groups in this class. Here, we prove results on the group <i>L</i> and prove that its genus is nontrivial.https://www.mdpi.com/2227-7390/11/12/2672mislin genusnoncancellationshort exact sequencepullback diagramlocalization |
| spellingShingle | Thandile Tonisi Rugare Kwashira Jules C. Mba On the Structure of the Mislin Genus of a Pullback Mathematics mislin genus noncancellation short exact sequence pullback diagram localization |
| title | On the Structure of the Mislin Genus of a Pullback |
| title_full | On the Structure of the Mislin Genus of a Pullback |
| title_fullStr | On the Structure of the Mislin Genus of a Pullback |
| title_full_unstemmed | On the Structure of the Mislin Genus of a Pullback |
| title_short | On the Structure of the Mislin Genus of a Pullback |
| title_sort | on the structure of the mislin genus of a pullback |
| topic | mislin genus noncancellation short exact sequence pullback diagram localization |
| url | https://www.mdpi.com/2227-7390/11/12/2672 |
| work_keys_str_mv | AT thandiletonisi onthestructureofthemislingenusofapullback AT rugarekwashira onthestructureofthemislingenusofapullback AT julescmba onthestructureofthemislingenusofapullback |