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<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {D}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">D</mi> </math></EquationSource> </InlineEquation> be a non-trivial design and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(G\le Aut(\mathcal {D})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>≤</mo> <mi>A</mi> <mi>u</mi> <mi>t</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">D</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be block-transitive. We give a sufficient and necessary condition that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathcal {D}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">D</mi> </math></EquationSource> </InlineEquation> is a <i>G</i>-block-transitive, (<i>G</i>,&#xa0;<i>s</i>)-chain imprimitive 3-design for any integer <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(s\ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, using this criterion, we find some examples of 3-designs with a <i>G</i>-block-transitive, (<i>G</i>,&#xa0;3)-chain imprimitive automorphism group.</p>

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On block-transitive 3-designs with a chain of imprimitive partitions

  • 期刊论文

  • 发表日期：

    2026年5月3日

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  • Yihui Li,
  • Shenglin Zhou

摘要

Let \(\mathcal {D}\) D be a non-trivial design and \(G\le Aut(\mathcal {D})\) G ≤ A u t ( D ) be block-transitive. We give a sufficient and necessary condition that \(\mathcal {D}\) D is a G-block-transitive, (G, s)-chain imprimitive 3-design for any integer \(s\ge 2\) s ≥ 2 . Furthermore, using this criterion, we find some examples of 3-designs with a G-block-transitive, (G, 3)-chain imprimitive automorphism group.