This paper contributes to the classification of flag-transitive 2-$ (v, k, \lambda) $ designs. Let $ \mathcal{D} $ be a non-trivial and non-symmetric $ 2 $-$ (v, k, \lambda) $ design with $ \lambda $ prime and $ G $ be a flag-transitive point-primitive automorphism group of $ \mathcal{D} $. A recent work by the first author and Chen has proven that the socle of $G$ is either a nonabelian simple group or an elementary abelian $ p $-group for some prime $ p $. In this paper, we focus on the case where the socle of $G$ is an exceptional group of Lie type and give all possible parameters of such 2-designs.
Citation: Yongli Zhang, Jiaxin Shen. Flag-transitive non-symmetric 2-designs with $ \lambda $ prime and exceptional groups of Lie type[J]. AIMS Mathematics, 2024, 9(9): 25636-25645. doi: 10.3934/math.20241252
This paper contributes to the classification of flag-transitive 2-$ (v, k, \lambda) $ designs. Let $ \mathcal{D} $ be a non-trivial and non-symmetric $ 2 $-$ (v, k, \lambda) $ design with $ \lambda $ prime and $ G $ be a flag-transitive point-primitive automorphism group of $ \mathcal{D} $. A recent work by the first author and Chen has proven that the socle of $G$ is either a nonabelian simple group or an elementary abelian $ p $-group for some prime $ p $. In this paper, we focus on the case where the socle of $G$ is an exceptional group of Lie type and give all possible parameters of such 2-designs.
[1] | S. H. Alavi, M. Bayat, A. Daneshkhah, Finite exceptional groups of Lie type and symmetric designs, Discrete Math., 345 (2022), 112894. https://doi.org/10.1016/j.disc.2022.112894 doi: 10.1016/j.disc.2022.112894 |
[2] | S. H. Alavi, Almost simple groups as flag-transitive automorphism groups of 2-designs with ${\lambda} = 2$, arXiv preprint, 2023. https://doi.org/10.48550/arXiv.2307.05195 |
[3] | M. Biliotti, A. Montinaro, Nonsymmetric 2-$(v, k, \lambda)$ designs, with $(r, \lambda) = 1$, admitting a solvable flag-transitive automorphism group of affine type, J. Comb. Des., 27 (2019), 784–800. https://doi.org/10.1002/jcd.21677 doi: 10.1002/jcd.21677 |
[4] | J. N. Bray, D. F. Holt, C. M. Roney-Dougal, The maximal subgroups of the low-dimensional finite classical groups, Cambridge University Press, 2013. https://doi.org/10.1017/cbo9781139192576 |
[5] | F. Buekenhout, A. Delandtsheer, J. Doyen, Finite linear spaces with flag-transitive groups, J. Comb. Theory A, 49 (1988), 268–293. https://doi.org/10.1016/0097-3165(88)90056-8 doi: 10.1016/0097-3165(88)90056-8 |
[6] | F. Buekenhout, A. Delandtsheer, J. Doyen, P. B. Kleidman, M. W. Liebeck, J. Saxl, Linear spaces with flag-transitive automorphism groups, Geometriae Dedicata, 36 (1990), 89–94. https://doi.org/10.1007/BF00181466 doi: 10.1007/BF00181466 |
[7] | H. Davies, Flag-transitivity and primitivity, Discrete Math., 63 (1987), 91–93. https://doi.org/10.1016/0012-365X(87)90154-3 doi: 10.1016/0012-365X(87)90154-3 |
[8] | P. Dembowski, Finite geometries, Springer-Verlag, New York, 1968. https://doi.org/10.1007/978-3-642-62012-6 |
[9] | B. Huppert, N. Blackburn, Finite groups III, Spring-Verlag, New York, 1982. https://doi.org/10.1007/978-3-642-67997-1 |
[10] | P. B. Kleidman, The finite flag-transitive linear spaces with an exceptional automorphism group, Finite Geometries and Combinatorial Designs (Lincoln, NE, 1987), 111 (1990), 117–136. https://doi.org/10.1090/conm/111/1079743 |
[11] | P. B. Kleidman, The maximal subgroups of the Chevalley groups $G_2(q)$ with $q$ odd, the Ree groups ${^2G_2(q)}$, and their automorphism groups, J. Algebra, 117 (1988), 30–71. https://doi.org/10.1016/0021-8693(88)90239-6 doi: 10.1016/0021-8693(88)90239-6 |
[12] | H. Li, Z. Zhang, S. Zhou, Flag-transitive automorphism groups of 2-designs with $\lambda>(r, \lambda)^2$ are not product type, J. Comb. Theory A, 208 (2024), 105923. https://doi.org/10.1016/j.jcta.2024.105923 doi: 10.1016/j.jcta.2024.105923 |
[13] | M. W. Liebeck, J. Saxl, The finite primitive permutation groups of rank three, B. Lond. Math. Soc., 18 (1986), 165–172. https://doi.org/10.1112/blms/18.2.165 doi: 10.1112/blms/18.2.165 |
[14] | M. W. Liebeck, J. Saxl, G. M. Seitz, On the overgroups of irreducible subgroups of the finite classical groups, P. Lond. Math. Soc., 3 (1987), 507–537. https://doi.org/10.1112/plms/s3-55.3.507 doi: 10.1112/plms/s3-55.3.507 |
[15] | H. L$\ddot{\mathrm{u}}$neburg, Some remarks concerning the Ree groups of type $(G2)$, J. Algebra, 3 (1966), 256–259. https://doi.org/10.1016/0021-8693(66)90014-7 doi: 10.1016/0021-8693(66)90014-7 |
[16] | A. Montinaro, M. Biliotti, E. Francot, Classification of the non-trivial 2-$(v, k, \lambda)$ designs, with $(r, \lambda) = 1$ and $\lambda>1$, admitting a non-solvable flag-transitive automorphism group of affine type, J. Algebr. Comb., 55 (2022), 853–889. https://doi.org/10.1007/s10801-021-01075-1 doi: 10.1007/s10801-021-01075-1 |
[17] | J. Saxl, On finite linear spaces with almost simple flag-transitive automorphism groups, J. Comb. Theory A, 100 (2002), 322–348. https://doi.org/10.1006/jcta.2002.3305 doi: 10.1006/jcta.2002.3305 |
[18] | G. M. Seitz, Flag-transitive subgroups of Chevalley groups, North-Holland Math. Stud., 7 (1973), 122–125. https://doi.org/10.1016/S0304-0208(08)71838-3 doi: 10.1016/S0304-0208(08)71838-3 |
[19] | M. Suzuki, On a class of doubly transitive groups, Ann. Math., 75 (1962), 105–145. https://doi.org/10.2307/1970423 doi: 10.2307/1970423 |
[20] | Y. Wang, S. Zhou, Symmetric designs admitting flag-transitive and point-primitive almost simple automorphism groups of Lie type, J. Algebra Appl., 16 (2017), 1750192. https://doi.org/10.1142/S0219498817501924 doi: 10.1142/S0219498817501924 |
[21] | X. Zhan, T, Zhou, S. Bai, S. Peng, L. Gan, Block-transitive automorphism groups on 2-designs with block size 4, Discrete Math., 343 (2020), 111726. https://doi.org/10.1016/j.disc.2019.111726 doi: 10.1016/j.disc.2019.111726 |
[22] | X. Zhan, S. Ding, A reduction for block-transitive triple systems, Discrete Math., 341 (2018), 2442–2447. https://doi.org/10.1016/j.disc.2018.05.021 doi: 10.1016/j.disc.2018.05.021 |
[23] | X. Zhang, S. Zhou, Block-transitive symmetric designs and alternating groups, Results Math., 78 (2023), 185. https://doi.org/10.1007/s00025-023-01964-w doi: 10.1007/s00025-023-01964-w |
[24] | X. Zhang, S. Zhou, Block-transitive and point-primitive 2-designs with sporadic socle, J. Comb. Des., 25 (2017), 231–238. https://doi.org/10.1002/jcd.21528 doi: 10.1002/jcd.21528 |
[25] | Y. Zhang, J. Chen, Reduction for flag-transitive point-primitive 2-$(v, k, \lambda)$ designs with $\lambda$ prime, J. Comb. Des., 32 (2024), 88–101. https://doi.org/10.1002/jcd.21927 doi: 10.1002/jcd.21927 |
[26] | Y. Zhang, S. Zhou, Flag-transitive non-symmetric 2-designs with $(r, \lambda) = 1$ and exceptional groups of lie type, Electron. J. Comb., 27 (2020), P2.9. https://doi.org/10.37236/8832 doi: 10.37236/8832 |
[27] | Y. Zhao, S. Zhou, Flag-transitive 2-$(v, k, \lambda)$ designs with $r>\lambda(k-3)$, Design. Code. Cryptogr., 90 (2022), 863–869. https://doi.org/10.1007/s10623-022-01010-w doi: 10.1007/s10623-022-01010-w |