On the group of unitriangular automorphisms of the polynomial ring in two variables over a finite field
The group \(U\!J_2(\mathbb{F}_q)\) of unitriangular automorphisms of the polynomial ring in two variables over a finite field \(\mathbb{F}_q\), \(q=p^m\), is studied. We proved that \(U\!J_2(\mathbb{F}_q)\) is isomorphic to a standard wreath product of elementary Abelian \(p\)-groups. Using wreath p...
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| Date: | 2018 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Lugansk National Taras Shevchenko University
2018
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| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1038 |
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| Journal Title: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543025739595776 |
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| author | Leshchenko, Yuriy Yu. Sushchansky, Vitaly I. |
| author_facet | Leshchenko, Yuriy Yu. Sushchansky, Vitaly I. |
| author_sort | Leshchenko, Yuriy Yu. |
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| collection | OJS |
| datestamp_date | 2018-04-26T02:11:00Z |
| description | The group \(U\!J_2(\mathbb{F}_q)\) of unitriangular automorphisms of the polynomial ring in two variables over a finite field \(\mathbb{F}_q\), \(q=p^m\), is studied. We proved that \(U\!J_2(\mathbb{F}_q)\) is isomorphic to a standard wreath product of elementary Abelian \(p\)-groups. Using wreath product representation we proved that the nilpotency class of \(U\!J_2(\mathbb{F}_q)\) is \(c=m(p-1)+1\) and the \((k+1)\)th term of the lower central series of this group coincides with the \((c-k)\)th term of its upper central series. Also we showed that \(U\!J_n(\mathbb{F}_q)\) is not nilpotent if \(n \geq 3\). |
| first_indexed | 2025-12-02T15:38:12Z |
| format | Article |
| id | admjournalluguniveduua-article-1038 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:38:12Z |
| publishDate | 2018 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-10382018-04-26T02:11:00Z On the group of unitriangular automorphisms of the polynomial ring in two variables over a finite field Leshchenko, Yuriy Yu. Sushchansky, Vitaly I. polynomial ring, unitriangular automorphism, finite field, wreath product, nilpotent group, central series 20D15, 20E22, 20E36, 20F14 The group \(U\!J_2(\mathbb{F}_q)\) of unitriangular automorphisms of the polynomial ring in two variables over a finite field \(\mathbb{F}_q\), \(q=p^m\), is studied. We proved that \(U\!J_2(\mathbb{F}_q)\) is isomorphic to a standard wreath product of elementary Abelian \(p\)-groups. Using wreath product representation we proved that the nilpotency class of \(U\!J_2(\mathbb{F}_q)\) is \(c=m(p-1)+1\) and the \((k+1)\)th term of the lower central series of this group coincides with the \((c-k)\)th term of its upper central series. Also we showed that \(U\!J_n(\mathbb{F}_q)\) is not nilpotent if \(n \geq 3\). Lugansk National Taras Shevchenko University 2018-04-26 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1038 Algebra and Discrete Mathematics; Vol 17, No 2 (2014) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1038/560 Copyright (c) 2018 Algebra and Discrete Mathematics |
| spellingShingle | polynomial ring unitriangular automorphism finite field wreath product nilpotent group central series 20D15 20E22 20E36 20F14 Leshchenko, Yuriy Yu. Sushchansky, Vitaly I. On the group of unitriangular automorphisms of the polynomial ring in two variables over a finite field |
| title | On the group of unitriangular automorphisms of the polynomial ring in two variables over a finite field |
| title_full | On the group of unitriangular automorphisms of the polynomial ring in two variables over a finite field |
| title_fullStr | On the group of unitriangular automorphisms of the polynomial ring in two variables over a finite field |
| title_full_unstemmed | On the group of unitriangular automorphisms of the polynomial ring in two variables over a finite field |
| title_short | On the group of unitriangular automorphisms of the polynomial ring in two variables over a finite field |
| title_sort | on the group of unitriangular automorphisms of the polynomial ring in two variables over a finite field |
| topic | polynomial ring unitriangular automorphism finite field wreath product nilpotent group central series 20D15 20E22 20E36 20F14 |
| topic_facet | polynomial ring unitriangular automorphism finite field wreath product nilpotent group central series 20D15 20E22 20E36 20F14 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1038 |
| work_keys_str_mv | AT leshchenkoyuriyyu onthegroupofunitriangularautomorphismsofthepolynomialringintwovariablesoverafinitefield AT sushchanskyvitalyi onthegroupofunitriangularautomorphismsofthepolynomialringintwovariablesoverafinitefield |