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

Document Type : Original Article


School of Mathematical and Computer Sciences, University of Limpopo (Turfloop), P Bag X1106, Sovenga 0727, South Africa


A finite group $G$ is said to be $(l,m, n)$-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 [Nova J. Algebra and Geometry, 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 nonabelian 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 $PSL(3,5),$ where we will determine the $(p,q,r)$-generations and the ranks of the classes of $PSL(3,5).$ 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.


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