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 $$T(l,m, n) = \left&lt;x, y, z|x^{l} = y^{m} = z^{n} = xyz = 1\right&gt;.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>l</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mfenced> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>z</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>l</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>y</mml:mi> <mml:mi>m</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>z</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>x</mml:mi> <mml:mi>y</mml:mi> <mml:mi>z</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mfenced> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> In [23], Moori posed the question of finding all the ( p , q , r ) triples, where $$p,\ q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mspace/> <mml:mi>q</mml:mi> </mml:mrow> </mml:math> 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 $$G_{2}(3).$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> We mainly used the structure constant method together with other results to establish the generation and non-generation of the $$G_{2}(3)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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.