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<p>A finite group <i>G</i> is called <i>uniformly semi-rational</i> if there exists an integer <i>r</i> such that the generators of every cyclic subgroup <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\langle x \rangle \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">⟨</mo> <mi>x</mi> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation> of <i>G</i> lie in at most two conjugacy classes, namely <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(x^G\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>x</mi> <mi>G</mi> </msup> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\((x^r)^G\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mi>r</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mi>G</mi> </msup> </math></EquationSource> </InlineEquation>. In this paper, we provide a classification of uniformly semi-rational non-abelian simple groups with particular focus on alternating groups.</p>

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Uniformly semi-rational simple groups

  • 期刊论文

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  • 发表日期：

    2025年5月6日

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  • Marco Vergani

摘要

A finite group G is called uniformly semi-rational if there exists an integer r such that the generators of every cyclic subgroup \(\langle x \rangle \) ⟨ x ⟩ of G lie in at most two conjugacy classes, namely \(x^G\) x G or \((x^r)^G\) ( x r ) G . In this paper, we provide a classification of uniformly semi-rational non-abelian simple groups with particular focus on alternating groups.