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Block-transitive 3-(v, k, 1) designs associated with alternating groups

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Abstract

Let \({\mathcal {D}}\) be a nontrivial 3-(v, k, 1) design admitting a block-transitive group G of automorphisms. A recent work of Gan and the second author asserts that G is either affine or almost simple. In this paper, it is proved that if G is almost simple with socle an alternating group, then \({\mathcal {D}}\) is the unique 3-(10, 4, 1) design, and \(G=\textrm{PGL}(2,9)\), \(\textrm{M}_{10}\) or \(\textrm{Aut}(\textrm{A}_6 )=\textrm{S}_6:\textrm{Z}_2\), and G is flag-transitive.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (12071484,12271524,12071023,11971054). The authors are very grateful for the anonymous referees’ valuable comments to improve the paper.

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Author notes

  1. Ting Lan, Weijun Liu, and Fu-Gang Yin have contributed equally to this work.

Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Yuelu South Road, Changsha, 410083, Hunan, People’s Republic of China

    Ting Lan, Weijun Liu & Fu-Gang Yin

  2. College of General Education, Guangdong University of Science and Technology, Dongguan, 523083, Guangdong, People’s Republic of China

    Weijun Liu

Authors
  1. Ting Lan
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  2. Weijun Liu
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  3. Fu-Gang Yin
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Correspondence to Fu-Gang Yin.

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Lan, T., Liu, W. & Yin, FG. Block-transitive 3-(v, k, 1) designs associated with alternating groups. Des. Codes Cryptogr. 91, 2791–2807 (2023). https://doi.org/10.1007/s10623-023-01215-7

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  • DOI: https://doi.org/10.1007/s10623-023-01215-7

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