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<p>Let <i>G</i> be a primitive permutation group acting on a finite set <i>X</i>. The orbital diameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textrm{diam}(X,G)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>diam</mtext> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is defined to be the supremum of the diameters of the (connected) orbital graphs of <i>G</i> after disregarding the directions of all edges in the graphs. This invariant is studied in the case when <i>G</i> is an almost simple group in a standard action. A lower bound is given for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\textrm{diam}(X,G)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>diam</mtext> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, and we provide a partial classification of pairs (<i>X</i>,&#xa0;<i>G</i>) for which the orbital diameter is at most 2.</p>

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On the orbital diameter of classical groups in standard actions

  • 期刊论文

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  • 发表日期：

    2026年7月23日

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  • Attila Maróti,
  • Kamilla Rekvényi

摘要

Let G be a primitive permutation group acting on a finite set X. The orbital diameter \(\textrm{diam}(X,G)\) diam ( X , G ) is defined to be the supremum of the diameters of the (connected) orbital graphs of G after disregarding the directions of all edges in the graphs. This invariant is studied in the case when G is an almost simple group in a standard action. A lower bound is given for \(\textrm{diam}(X,G)\) diam ( X , G ) , and we provide a partial classification of pairs (X, G) for which the orbital diameter is at most 2.