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<p>A Steiner triple system STS(<i>v</i>) is called <i>f</i>-pyramidal if it has an automorphism group fixing <i>f</i> points and acting sharply transitively on the remaining <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(v-f\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>v</mi> <mo>-</mo> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation> points. In this paper, we focus on the STSs that are <i>f</i>-pyramidal over some abelian group. Their existence has been settled only for the smallest admissible values of <i>f</i>, that is, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(f=0,1,3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>.</p><p>In this paper, we complete this result and determine, for every <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(f&gt;3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>&gt;</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, the spectrum of values (<i>f</i>,&#xa0;<i>v</i>) for which there is an <i>f</i>-pyramidal STS(<i>v</i>) over an abelian group. This result is obtained by constructing difference families relative to a suitable partial spread.</p>

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The existence of pyramidal Steiner triple systems over abelian groups

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月28日

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  • Yanxun Chang,
  • Tommaso Traetta,
  • Junling Zhou

摘要

A Steiner triple system STS(v) is called f-pyramidal if it has an automorphism group fixing f points and acting sharply transitively on the remaining \(v-f\) v - f points. In this paper, we focus on the STSs that are f-pyramidal over some abelian group. Their existence has been settled only for the smallest admissible values of f, that is, \(f=0,1,3\) f = 0 , 1 , 3 .

In this paper, we complete this result and determine, for every \(f>3\) f > 3 , the spectrum of values (f, v) for which there is an f-pyramidal STS(v) over an abelian group. This result is obtained by constructing difference families relative to a suitable partial spread.