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Block-Transitive Symmetric Designs and Alternating Groups

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This paper studies symmetric \(\varvec{(v,k,\lambda )}\) designs \(\varvec{{\mathcal {D}}}\) with \(\varvec{v}\) odd and admitting a block-transitive automorphism group \(\varvec{G}\) whose socle is an alternating group \(\varvec{\mathrm {A_{n}}}\). We indeed show that for given \(\varvec{\lambda }\), only finitely many such designs exist and prove that if \(\varvec{2\le \lambda \le 5}\), then \(\varvec{{\mathcal {D}}=\mathrm {PG_{2}(3,2)}}\) or its complement and \(\varvec{G=\mathrm {S_{5}}, \mathrm {S_{6}}, \mathrm {A_{n}}}\) \(\varvec{(n=5,6,7,8)}\).

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Funding

**aohong Zhang is supported by the Natural Science Foundation of Shanxi Province, China (Grant No.202103021223085) and Shenglin Zhou is supported by the National Natural Science Foundation of China (Grant No.12271173).

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Authors and Affiliations

  1. School of Mathematics, Taiyuan University of Technology, Taiyuan, 030024, People’s Republic of China

    **aohong Zhang

  2. School of Mathematics, South China University of Technology, Guangzhou, 510641, People’s Republic of China

    Shenglin Zhou

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  1. **aohong Zhang
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  2. Shenglin Zhou
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Correspondence to **aohong Zhang.

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Zhang, X., Zhou, S. Block-Transitive Symmetric Designs and Alternating Groups. Results Math 78, 185 (2023). https://doi.org/10.1007/s00025-023-01964-w

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