The (p, q, r)-generations of the Chevalley group \(G_{2}(3)\)

Abstract

A finite group G is called (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 [23], 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 a (p, q, r)-generated. In answering this question, we establish all the (p, q, r)-generations for the group \(G_{2}(3).\) We mainly used the structure constant method together with other results to establish the generation and non-generation of the \(G_{2}(3)\) by the triples (p, q, r). The Groups, Algorithms and Programming, GAP [21] and the Atlas of finite group representations [27] are used in our computations.