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<p>We construct a primitive permutation action of the Steinberg triality group <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="MediaObjects/13_2026_2240_IEq1_HTML.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="120" Type="Linedraw" Width="49" /> </InlineMediaObject> </InlineEquation> of degree 4,&#xa0;064,&#xa0;256 and show that there are distinct points <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\alpha ,\beta \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> </math></EquationSource> </InlineEquation> such that there is no derangement <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="MediaObjects/13_2026_2240_IEq3_HTML.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="120" Type="Linedraw" Width="79" /> </InlineMediaObject> </InlineEquation> with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\alpha ^g=\beta \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>α</mi> <mi>g</mi> </msup> <mo>=</mo> <mi>β</mi> </mrow> </math></EquationSource> </InlineEquation>. This answers a question by John G.&#xa0;Thompson (Problem 8.75 in the Kourovka Notebook) in the negative.</p>

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On transitive sets of derangements in primitive groups

  • 期刊论文

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  • 发表日期：

    2026年4月20日

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  • Peter Müller

摘要

We construct a primitive permutation action of the Steinberg triality group of degree 4, 064, 256 and show that there are distinct points \(\alpha ,\beta \) α , β such that there is no derangement with \(\alpha ^g=\beta \) α g = β . This answers a question by John G. Thompson (Problem 8.75 in the Kourovka Notebook) in the negative.