A finite group G is called (l, m, n)-generated, if it is a quotient group of the triangle group <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$T(l,m, n) = \left<x, y, z|x^{l} = y^{m} = z^{n} = xyz = 1\right>.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mrow> <mo stretchy="false">(</mo> <mi>l</mi> <mo>,</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mfenced close="〉" open="〈"> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">|</mo> </mrow> <msup> <mi>x</mi> <mi>l</mi> </msup> <mo>=</mo> <msup> <mi>y</mi> <mi>m</mi> </msup> <mo>=</mo> <msup> <mi>z</mi> <mi>n</mi> </msup> <mo>=</mo> <mi>x</mi> <mi>y</mi> <mi>z</mi> <mo>=</mo> <mn>1</mn> </mfenced> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> In [<CitationRef CitationID="CR23">23</CitationRef>], Moori posed the question of finding all the (p, q, r) triples, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p,\ q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>,</mo> <mspace width="4pt" /> <mi>q</mi> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$G_{2}(3).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>G</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> We mainly used the structure constant method together with other results to establish the generation and non-generation of the <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$G_{2}(3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>G</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> by the triples (p, q, r). The Groups, Algorithms and Programming, GAP [<CitationRef CitationID="CR21">21</CitationRef>] and the Atlas of finite group representations [<CitationRef CitationID="CR27">27</CitationRef>] are used in our computations.

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

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2026年3月4日

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A. B. M. Basheer,
M. J. Motalane,
M. G. Sehoana,
T. T. Seretlo

摘要

A finite group G is called (l, m, n)-generated, if it is a quotient group of the triangle group $T(l,m, n) = \left<x, y, z|x^{l} = y^{m} = z^{n} = xyz = 1\right>.$ $T (l, m, n) = 〈x, y, z | x^{l} = y^{m} = z^{n} = x y z = 1〉 .$ In [23], Moori posed the question of finding all the (p, q, r) triples, where $p,\ q$ $p, q$ 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 $G_{2}(3).$ $G_{2} (3) .$ We mainly used the structure constant method together with other results to establish the generation and non-generation of the $G_{2}(3)$ $G_{2} (3)$ 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.

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

摘要