On two generation methods for the simple linear group $PSL(3,7)$
A finite group $G$ is said to be \textit{$(l,m, n)$-generated}, if it is a quotient group of the triangle group $T(l,m, n) = \left<x, y, z|x^{l} = y^{m} = z^{n} = xyz = 1\right>.$ In [J. Moori, $(p, q, r)$-generations for the Janko groups $J_{1}$ and $J_{2}$, Nova J. Algebra and Geometry, \bf{...
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| Format: | Article |
| Language: | English |
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Amirkabir University of Technology
2023-02-01
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| Series: | AUT Journal of Mathematics and Computing |
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| Online Access: | https://ajmc.aut.ac.ir/article_4929_a2b1bd2dc7f610ad4ece8144155d1ced.pdf |
| _version_ | 1797307062704668672 |
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| author | Thekiso Seretlo |
| author_facet | Thekiso Seretlo |
| author_sort | Thekiso Seretlo |
| collection | DOAJ |
| description | A finite group $G$ is said to be \textit{$(l,m, n)$-generated}, if it is a quotient group of the triangle group $T(l,m, n) = \left<x, y, z|x^{l} = y^{m} = z^{n} = xyz = 1\right>.$ In [J. Moori, $(p, q, r)$-generations for the Janko groups $J_{1}$ and $J_{2}$, Nova J. Algebra and Geometry, \bf{2} (1993), no. 3, 277--285], Moori posed the question of finding all the $(p,q,r)$ triples, where $p,\ q$ and $r$ are prime numbers, such that a non-abelian finite simple group $G$ is $(p,q,r)$-generated. Also for a finite simple group $G$ and a conjugacy class $X$ of $G,$ the rank of $X$ in $G$ is defined to be the minimal number of elements of $X$ generating $G.$ In this paper we investigate these two generational problems for the group $PSL(3,7),$ where we will determine the $(p,q,r)$-generations and the ranks of the classes of $PSL(3,7).$ We approach these kind of generations using the structure constant method. GAP [The GAP Group, GAP -- Groups, Algorithms, and Programming, Version 4.9.3; 2018. (http://www.gap-system.org)] is used in our computations. |
| first_indexed | 2024-03-08T00:51:53Z |
| format | Article |
| id | doaj.art-612b42cddc584d48a8bfaff5cccb237f |
| institution | Directory Open Access Journal |
| issn | 2783-2449 2783-2287 |
| language | English |
| last_indexed | 2024-03-08T00:51:53Z |
| publishDate | 2023-02-01 |
| publisher | Amirkabir University of Technology |
| record_format | Article |
| series | AUT Journal of Mathematics and Computing |
| spelling | doaj.art-612b42cddc584d48a8bfaff5cccb237f2024-02-14T19:39:05ZengAmirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24492783-22872023-02-0141273710.22060/ajmc.2022.21638.10954929On two generation methods for the simple linear group $PSL(3,7)$Thekiso Seretlo0School of Mathematical Sciences, North West University, Mafikeng Branch P/B X2046, Mmabatho 2735, South AfricaA finite group $G$ is said to be \textit{$(l,m, n)$-generated}, if it is a quotient group of the triangle group $T(l,m, n) = \left<x, y, z|x^{l} = y^{m} = z^{n} = xyz = 1\right>.$ In [J. Moori, $(p, q, r)$-generations for the Janko groups $J_{1}$ and $J_{2}$, Nova J. Algebra and Geometry, \bf{2} (1993), no. 3, 277--285], Moori posed the question of finding all the $(p,q,r)$ triples, where $p,\ q$ and $r$ are prime numbers, such that a non-abelian finite simple group $G$ is $(p,q,r)$-generated. Also for a finite simple group $G$ and a conjugacy class $X$ of $G,$ the rank of $X$ in $G$ is defined to be the minimal number of elements of $X$ generating $G.$ In this paper we investigate these two generational problems for the group $PSL(3,7),$ where we will determine the $(p,q,r)$-generations and the ranks of the classes of $PSL(3,7).$ We approach these kind of generations using the structure constant method. GAP [The GAP Group, GAP -- Groups, Algorithms, and Programming, Version 4.9.3; 2018. (http://www.gap-system.org)] is used in our computations.https://ajmc.aut.ac.ir/article_4929_a2b1bd2dc7f610ad4ece8144155d1ced.pdfconjugacy classes$(p,q,r)$-generationrankstructure constant |
| spellingShingle | Thekiso Seretlo On two generation methods for the simple linear group $PSL(3,7)$ AUT Journal of Mathematics and Computing conjugacy classes $(p,q,r)$-generation rank structure constant |
| title | On two generation methods for the simple linear group $PSL(3,7)$ |
| title_full | On two generation methods for the simple linear group $PSL(3,7)$ |
| title_fullStr | On two generation methods for the simple linear group $PSL(3,7)$ |
| title_full_unstemmed | On two generation methods for the simple linear group $PSL(3,7)$ |
| title_short | On two generation methods for the simple linear group $PSL(3,7)$ |
| title_sort | on two generation methods for the simple linear group psl 3 7 |
| topic | conjugacy classes $(p,q,r)$-generation rank structure constant |
| url | https://ajmc.aut.ac.ir/article_4929_a2b1bd2dc7f610ad4ece8144155d1ced.pdf |
| work_keys_str_mv | AT thekisoseretlo ontwogenerationmethodsforthesimplelineargrouppsl37 |