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Block Designs with \(\gcd (r,\lambda )=1\) Admitting Flag-Transitive Automorphism Groups

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Abstract

In this paper, we present a classification of \(\mathbf{2}\)-designs with \(\mathbf{gcd}({{\varvec{r}}},{\varvec{\lambda }})=\mathbf{1}\) admitting flag-transitive automorphism groups. If \({{\varvec{G}}}\) is a flag-transitive automorphism group of a non-trivial \(\mathbf{2}\)-design \(\mathcal {{\varvec{D}}}\) with \(\mathbf{gcd}({{\varvec{r}}},{\varvec{\lambda }})=\mathbf{1}\), then either \((\mathcal {{\varvec{D}}},{{\varvec{G}}})\) is one of the known examples described in this paper, or \(\mathcal {{\varvec{D}}}\) has \({{\varvec{q}}} = {{\varvec{p}}}^{{{\varvec{d}}}}\) points with \({{\varvec{p}}}\) prime and \({{\varvec{G}}}\) is a subgroup of \(\mathbf{A}{\varvec{\Gamma }}{} \mathbf{L}_{\mathbf{1}}({{\varvec{q}}})\).

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Funding

Shenglin Zhou is supported by the National Natural Science Foundation of China (Grant No.11871224).

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Bu-Ali Sina University, Hamedan, Iran

    Seyed Hassan Alavi, Mohsen Bayat, Ashraf Daneshkhah & Fatemeh Mouseli

  2. Dipartimento di Matematica e Fisica “E. De Giorgi”, University of Salento, Lecce, Italy

    Mauro Biliotti, Eliana Francot, Alessandro Montinaro & Pierluigi Rizzo

  3. Department of Mathematics, China Three Gorges University, Yichang, Hubei,  443002, People’s Republic of China

    Haiyan Guan

  4. Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong,  510303, People’s Republic of China

    Delu Tian

  5. School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou, Henan,  450046, People’s Republic of China

    Yajie Wang

  6. School of Science, East China JiaoTong University, Nanchang, Jiangxi,  330013, People’s Republic of China

    Xiaoqin Zhan

  7. Department of Mathematics, Guangdong Polytechnic Normal University, Guangzhou, Guangdong, v510665, People’s Republic of China

    Yongli Zhang

  8. School of Mathematics, South China University of Technology, Guangzhou, Guangdong,  510640, People’s Republic of China

    Shenglin Zhou

  9. School of Medical Information and Engineering, Xuzhou Medical University, Xuzhou, Jiangsu,  221004, People’s Republic of China

    Yan Zhu

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  1. Seyed Hassan Alavi
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Correspondence to Ashraf Daneshkhah.

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Alavi, S.H., Bayat, M., Biliotti, M. et al. Block Designs with \(\gcd (r,\lambda )=1\) Admitting Flag-Transitive Automorphism Groups. Results Math 77, 151 (2022). https://doi.org/10.1007/s00025-022-01697-2

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