On Two Generation Methods for The Simple Linear Group $PSL(3,5)$

Abstract

A finite group $G$ is said to be $(l,m, n)$-generated, if\xa0it is a quotient group of the triangle group $T(l,m, n) = left .$ In [Nova J. Algebra and Geometry, 2 (1993), no. 3,\xa0277--285], Moori posed the question of finding all the $(p,q,r)$\xa0triples, where $p, q,$ and $r$ are prime numbers, such that a\xa0nonabelian finite simple group $G$ is $(p,q,r)$-generated. Also for\xa0a 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\xa0elements of $X$ generating $G.$ In this paper, we investigate these\xa0two generational problems for the group $PSL(3,5),$ where we will\xa0determine the $(p,q,r)$-generations and the ranks of the classes of\xa0$PSL(3,5).$ We approach these kind of generations using the\xa0structure constant method. GAP [The GAP Group, GAP-Groups, Algorithms, and Programming, Version 4.9.3; 2018.\xa0(http://www.gap-system.org)] is used in our computations.