On two generation methods for the simple linear group $PSL(3,7)$

Document Type : Original Article


School of Mathematical Sciences, North West University, Mafikeng Branch P/B X2046, Mmabatho 2735, South Africa


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.


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