@article { author = {Seretlo, Thekiso}, title = {On two generation methods for the simple linear group $PSL(3,7)$}, journal = {AUT Journal of Mathematics and Computing}, volume = {4}, number = {1}, pages = {27-37}, year = {2023}, publisher = {Amirkabir University of Technology}, issn = {2783-2449}, eissn = {2783-2287}, doi = {10.22060/ajmc.2022.21638.1095}, abstract = {A finite group $G$ is said to be \textit{$(l,m, n)$-generated}, if it is a quotient group of the triangle group $T(l,m, n) = \left.$ 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.}, keywords = {conjugacy classes,$(p,q,r)$-Generation,rank,structure constant}, url = {https://ajmc.aut.ac.ir/article_4929.html}, eprint = {https://ajmc.aut.ac.ir/article_4929_a2b1bd2dc7f610ad4ece8144155d1ced.pdf} }