Abstract
Let \({\mathcal{D}}\) be a nontrivial triplane, and G be a subgroup of the full automorphism group of \({\mathcal{D}}\). In this paper we prove that if \({\mathcal{D}}\) is a triplane, \({G\leq Aut(\mathcal{D})}\) is flag-transitive, point-primitive and Soc(G) is an alternating group, then \({\mathcal{D}}\) is the projective space PG 2(3, 2), and \({G\cong A_7}\) with the point stabiliser \({G_x\cong PSL_3(2)}\).
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References
Aschbacher M.: On collineation groups of symmetric block designs. J. Comb. Theory Ser. A 11, 272–281 (1971)
Conway J.H., Curtis R.T., Norton S.P., Parker R.A., Wilson R.A.: Atlas of Finite Groups. Oxford University Press, London (1985)
Delandtsheer A.: Finite flag-transitive linear spaces with alternating socle. Algebraic Combinatorics and Applications (Gößeinstein, 1999), Springer, Berlin, 79–88 (2001).
Dembowski P.: Finite Geometries. Springer, Berlin (1968)
Dixon J.D., Mortimer B.: Permutation Groups. Springer-Verlag (1996).
Huppert B.: Endliche Gruppen I. Springer, Berlin (1967)
Kantor W.M.: Classification of 2-transitive symmetric designs. Graph Comb. 1, 165–166 (1985)
Liebeck M.W., Saxl J.: The primitive permutation groups of odd degree. J. Lond. Math. Soc.(2) 31, 250–264 (1985)
Liebeck M.W., Praeger C.E., Saxl J.: On the O’Nan-Scott theorem for finite primitive permutation groups. J. Aust. Math. Soc. Ser. A 44, 389–396 (1988)
Praeger C.E.: The flag-transitive symmetric designs with 45 points, blocks of size 12, and 3 blocks on every point pair. Des. Codes Cryptogr. 44, 115–132 (2007)
Praeger C.E., Zhou S.L.: Imprimitive flag-transitive symmetric designs. J. Comb. Theory Ser. A 113(7), 1381–1395 (2006)
Regueiro E.O’R.: On primitivity and reduction for flag-transitive symmetric designs. J. Comb. Theory Ser. A 109(1), 135–148 (2005)
Regueiro E.O’R.: Biplanes with flag-transitive automorphism groups of almost simple type, with alternating or sporadic socle. Eur. J. Comb. 26, 577–584 (2005)
Regueiro E.O’R.: Biplanes with flag-transitive automorphism groups of almost simple type, with classical socle. J. Algebr. Comb. 26, 529–552 (2007)
Regueiro E.O’R.: Biplanes with flag-transitive automorphism groups of almost simple type, with exceptional socle of Lie type. J. Algebr. Comb. 27, 479–491 (2008)
Zhou S.L., Dong H.L.: Sporadic groups and flag-transitive triplanes. Sci. China Ser. A 52(2), 394–400 (2009)
Zhou S.L., Dong H.L.: Exceptional groups of Lie type and flag-transitive triplanes, Sci. China Ser. A 52 (2009). doi:10.1007/s11425-009-0051-5.
Zhou S.L., Dong H.L.: Finite classical groups and flag-transitive triplanes. Discrete Math. 309(16), 5183–5195 (2009)
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Communicated by J.D. Key.
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Zhou, S., Dong, H. Alternating groups and flag-transitive triplanes. Des. Codes Cryptogr. 57, 117–126 (2010). https://doi.org/10.1007/s10623-009-9355-2
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DOI: https://doi.org/10.1007/s10623-009-9355-2