Flag-transitive, point-imprimitive 2-\((v,k,\lambda )\) symmetric designs with k and \(\lambda \) prime powers

Abstract

Let \({\mathcal {D}}\) be a 2-\((v,k,\lambda )\) symmetric design with k and \(\lambda \) prime powers. If \({\mathcal {D}}\) admits a flag-transitive, point-imprimitive automorphism group G, we show that both k and \(\lambda \) must be powers of 2. Moreover, there exists an integer m such that either (a) \({\mathcal {D}}\) has parameters \((v,k,\lambda )=(2^{2m+2}-1, 2^{2m+1}, 2^{2m})\), and G preserves a partition of the points into \(2^{m+1}+1\) classes of size \(2^{m+1}-1\), or (b) \({\mathcal {D}}\) has parameters \((v,k,\lambda )=((2^{2m-1}-2^m+1)(2^{m-1}+1),\ 2^{2m-1},\ 2^m)\), and G preserves a partition of the points into \(2^{2m-1}-2^m+1\) classes of size \(2^{m-1}+1\).

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Acknowledgements

The authors thank the anonymous referees for their valuable suggestions and comments which helped to improve the paper.

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Correspondence to Shenglin Zhou.

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This work is supported by the National Natural Science Foundation of China (No.11871224).

Communicated by C. E. Praeger.

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Chen, J., Zhou, S. Flag-transitive, point-imprimitive 2-\((v,k,\lambda )\) symmetric designs with k and \(\lambda \) prime powers. Des. Codes Cryptogr. (2021). https://doi.org/10.1007/s10623-021-00869-5

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Keywords

  • Symmetric design
  • Automorphism group
  • Flag-transitive
  • Primitivity

Mathematics Subject Classification

  • 05B05
  • 05B25
  • 20B25