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Block-transitive 3-(v, k, 1) designs on exceptional groups of Lie type

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Abstract

Let \({\mathcal {D}}\) be a non-trivial G-block-transitive 3-(v, k, 1) design, where \(T\le G \le \textrm{Aut}(T)\) for some finite non-abelian simple group T. It is proved that if T is a simple exceptional group of Lie type, then T is either the Suzuki group \({}^2B_2(q)\) or \(G_2(q)\). Furthermore, if \(T={}^2B_2(q)\) then the design \({\mathcal {D}}\) has parameters \(v=q^2+1\) and \(k=q+1\), and so \({\mathcal {D}}\) is an inverse plane of order q, and if \(T=G_2(q)\) then the point stabilizer in T is either \(\textrm{SL}_3(q).2\) or \(\textrm{SU}_3(q).2\), and the parameter k satisfies very restricted conditions.

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Acknowledgements

We are grateful for the anonymous referees’ comments to improve the paper. We acknowledge the algebra software package Magma [4] for computation. Weijun Liu was supported by the National Natural Science Foundation of China (12071484, 12271524); Fu-Gang Yin was supported by the National Natural Science Foundation of China (12301461,12331013).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, 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

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  1. Ting Lan
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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 on exceptional groups of Lie type. J Algebr Comb (2024). https://doi.org/10.1007/s10801-024-01315-0

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