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 (l,m, n)-generated, if it is a quotient group of the triangle group T(l,m, n) = ⟨x, y, z| x l =y m = z n= xyz = 1⟩. In [J. Moori, (p, q, r)-generations for the Janko groups J1 and J2, 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 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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