Flag-transitive 2-\upsilon ,\mathrm{5},\lambda designs with sporadic socle

Abstract

We state that the ag-transitive automorphism group of a 2-\upsilon ,\mathrm{5},\lambda design \mathcal{D} is primitive of affine type or almost simple type. We also find that there are up to isomorphism 20 2-\upsilon ,\mathrm{5},\lambda designs admitting flag-transitive automorphism groups with socle of a sporadic simple group.

Keywords

2-design / primitivity / flag-transitivity / sporadic simple group

References

[1] Biggs N, White A. Permutation Groups and Combinatorial Structures. Cambridge: Cambridge Univ Press, 1979

[2] Bray N, Wilson A. Explicit representations of the maximal subgroups of the Monster. J Algebra, 2006, 300: 834–857

[3] Buekenhout F, Delandtsheer A, Doyen J. Finite linear spaces with flag-transitive groups. J Combin Theory Ser A, 1988, 49: 268–293

[4] Buekenhout F, Delandtsheer A, Doyen J, Kleidman P, Liebeck M, Saxl J. Linear spaces with flag-transitive automorphism groups. Geom Dedicata, 1990, 36: 89–94

[5] Cameron P. Finite permutation groups and finite simple groups. Bull Lond Math Soc, 1981, 13: 1–22

[6] Camina A, Mischke S. Line-transitive automorphism groups of linear spaces. Electron J Combin, 1996, 3: #R3

[7] Colbourn C, Dinitz J. The CRC Handbook of Combinatorial Designs. Boca Raton: CRC Press, 2007

[8] Conway J, Curtis R, Norton S, Parker R, Wilson R. Atlas of Finite Groups. Eynsham: Oxford Univ Press, 1985

[9] Davies D. Automorphisms of Designs. Ph D Thesis, University of East Anglia, 1987

[10] Dixon J, Mortimer B. Permutation Groups. New York: Springer-Verlag, 1996

[11] Liang H, Zhou S. Flag-transitive point-primitive automorphism groups of non-symmetric 2-(v, k, 2) designs. J Combin Des, 2016, 24: 421–435

[12] Liebeck M, Praeger C, Saxl J. On the O'Nan-Scott theorem for finite primitive permutation groups. J Aust Math Soc, 1988, 44: 389–396

[13] O’Relly Regueiro E. On primitivity and reduction for flag-transitive symmetric designs. J Combin Theory Ser A, 2005, 109: 135–148

[14] The GAP Group. GAP–Groups, Algorithms, and Programming, Version 4.7.4. 2014

[15] Tian D, Zhou S. Flag-transitive point-primitive symmetric (v, k, λ) designs with λ at most 100. J Combin Des, 2013, 21: 127–141

[16] Wilson R. The Finite Simple Groups. London: Springer-Verlag, 2009

[17] Zhan X, Ding S. A reduction for block-transitive triple systems. Discrete Math, 2018, 341: 2442–2447

[18] Zhan X, Zhou S, Chen G. Flag-transitive 2-(v, 4, λ) designs of product type. J Combin Des, 2018, 28: 445{462

[19] Zieschang P. Flag-transitive automorphism groups of 2-designs with (r, λ) = 1. J Algebra, 1988, 118: 265–275

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