<p>In this article, we investigate nontrivial symmetric <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(v,k,\lambda )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> designs with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\lambda \leqslant 10$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>⩽</mo> <mn>10</mn> </mrow> </math></EquationSource> </InlineEquation> admitting a flag-transitive and point-primitive affine automorphism group. In conclusion, all symmetric <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$(v,k,\lambda )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> designs admitting flag-transitive automorphism groups are known apart from those admitting one-dimensional automorphisms, and hence the classification of flag-transitive symmetric <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$(v,k,\lambda )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> designs with <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\lambda \leqslant 10$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>⩽</mo> <mn>10</mn> </mrow> </math></EquationSource> </InlineEquation> reduces to the case of one-dimensional affine automorphism groups.</p>

On flag-transitive and point-primitive symmetric $(v,k,\lambda )$ designs of affine type with $\lambda \leqslant 10$

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2025年10月25日

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Seyed Hassan Alavi,
Ashraf Daneshkhah,
Alessandro Montinaro

摘要

In this article, we investigate nontrivial symmetric $(v,k,\lambda )$ $(v, k, λ)$ designs with $\lambda \leqslant 10$ $λ ⩽ 10$ admitting a flag-transitive and point-primitive affine automorphism group. In conclusion, all symmetric $(v,k,\lambda )$ $(v, k, λ)$ designs admitting flag-transitive automorphism groups are known apart from those admitting one-dimensional automorphisms, and hence the classification of flag-transitive symmetric $(v,k,\lambda )$ $(v, k, λ)$ designs with $\lambda \leqslant 10$ $λ ⩽ 10$ reduces to the case of one-dimensional affine automorphism groups.

On flag-transitive and point-primitive symmetric \((v,k,\lambda )\) designs of affine type with \(\lambda \leqslant 10\)

摘要