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Designs, Codes and Cryptography

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15-05-2023

Block-transitive 3-(v, k, 1) designs associated with alternating groups

Authors: Ting Lan, Weijun Liu, Fu-Gang Yin

Published in: Designs, Codes and Cryptography | Issue 8/2023

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Abstract

Let \({\mathcal {D}}\) be a nontrivial 3-(v, k, 1) design admitting a block-transitive group G of automorphisms. A recent work of Gan and the second author asserts that G is either affine or almost simple. In this paper, it is proved that if G is almost simple with socle an alternating group, then \({\mathcal {D}}\) is the unique 3-(10, 4, 1) design, and \(G=\textrm{PGL}(2,9)\), \(\textrm{M}_{10}\) or \(\textrm{Aut}(\textrm{A}_6 )=\textrm{S}_6:\textrm{Z}_2\), and G is flag-transitive.

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Block R.E.: On the orbits of collineation group. Math. Z. 96, 33–49 (1967).MathSciNetCrossRefMATH

Bosma W., Cannon J., Playoust C.: The MAGMA algebra system I: the user language. J. Symb. Comput. 24, 235–265 (1997).MathSciNetCrossRefMATH

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

Cameron P.J.: Parallelisms of Complete Designs. Cambridge University Press, Cambridge (1976).CrossRefMATH

Cameron P.J., Praeger C.E.: Block-transitive \(t\)-designs I: point-imprimitive designs. Discret. Math. 118, 33–43 (1993).MathSciNetCrossRefMATH

Cameron P.J., Praeger C.E.: Block-transitive \(t\)-designs, II: large \(t\). In: Clerck F., Hirschfeld J. (eds.) Finite Geometry and Combinatorics, pp. 103–119. Cambridge University Press, Cambridge (1993).CrossRef

Camina A.R., Praeger C.E.: Line-transitive, point quasiprimitive automorphism groups of finite linear spaces are affine or almost simple. Aequationes Math. 61, 221–232 (2001).MathSciNetCrossRefMATH

Camina A., Spiezia F.: Sporadic groups and automorphisms of linear spaces. J. Comb. Des. 8, 353–362 (2000).MathSciNetCrossRefMATH

Camina A.R., Neumann P.M., Praeger C.E.: Alternating groups acting on finite linear spaces. Proc. Lond. Math. Soc. 87, 29–53 (2003).MathSciNetCrossRefMATH

10.

Camina A.R., Gill N., Zalesski A.E.: Large dimensional classical groups and linear spaces. Bull. Belg. Math. Soc. Simon Stevin. 15, 705–731 (2008).MathSciNetCrossRefMATH

11.

Conway J.H., Curtis R.T., Norton S.P., Parker R.A., Wilson R.A.: Atlas of Finite Groups. Oxford University Press, New York (1985).MATH

12.

Dixon J.D., Mortimer B.: Permutation Groups. Springer, New York (1996).CrossRefMATH

13.

Gan Y.S., Liu W.J.: Block-transitive point-primitive automorphism groups of Steiner \(3\)-designs. arXiv:2112.00466v3

14.

Gill N.: \(PSL(3, q)\) and line-transitive linear spaces. Beitr. Algebra Geom. 48, 591–620 (2007).MathSciNetMATH

15.

Huber M.: The classification of flag-transitive Steiner \(3\)-designs. Adv. Geom. 5, 195–221 (2005).MathSciNetCrossRefMATH

16.

Huber M.: The classification of flag-transitive Steiner \(4\)-designs. J. Algebr. Comb. 26, 183–207 (2007).MathSciNetCrossRefMATH

17.

Huber M.: The classification of flag-transitive Steiner \(5\)-designs. In: Huber M. (ed.) Flag-Transitive Steiner Designs, pp. 93–109. Birkhäuser, Basel (2009).CrossRefMATH

18.

Huber M.: Block-transitive designs in affine spaces designs. Des. Codes Cryptogr. 55, 235–242 (2010).MathSciNetCrossRefMATH

19.

Huber M.: On the Cameron-Praeger conjecture. J. Comb. Theory Ser. A. 117, 196–203 (2010).MathSciNetCrossRefMATH

20.

Huber M.: On the existence of block-transitive combinatorial designs. Discret. Math. Theor. Comput. Sci. 12, 123–132 (2010).MathSciNetMATH

21.

Kantor W.M.: \(k\)-homogeneous groups. Math. Z. 124, 261–265 (1972).MathSciNetCrossRefMATH

22.

Kantor W.M.: Homogeneous designs and geometric lattices. J. Comb. Theory Ser. A. 38, 66–74 (1985).MathSciNetCrossRefMATH

23.

Kantor W.M.: Primitive permutation groups of odd degree, and an application to finite projective planes. J. Algebra 106, 15–45 (1987).MathSciNetCrossRefMATH

24.

Liebeck M.W., Praeger C.E., Saxl J.: A classification of the maximal subgroups of the finite alternating and symmetric groups. J. Algebra 111, 365–383 (1987).MathSciNetCrossRefMATH

25.

Liu W.J.: The Chevalley groups \(G_2(2^n)\) and \(2\)-\((v, k, 1)\) designs. Algebra Colloq. 8, 471–480 (2001).MathSciNetMATH

26.

Liu W.J.: Finite linear spaces admitting a projective group \(PSU(3, q)\) with \(q\) even. Linear Algebra Appl. 374, 291–305 (2003).MathSciNetCrossRefMATH

27.

Liu W.J.: Finite linear spaces admitting a two-dimensional projective linear group. J. Comb. Theory Ser. A. 103, 209–222 (2003).MathSciNetCrossRefMATH

28.

Liu W.J.: The Chevalley groups \(G_2(q)\) with \(q\) odd and \(2\)-\((v, k,1)\) designs. Eur. J. Comb. 24, 331–346 (2003).CrossRefMATH

29.

Liu W.J., Li H.L., Ma C.G.: Suzuki groups and \(2\)-\((v, k, 1)\) designs. Eur. J. Comb. 22, 513–519 (2001).MathSciNetCrossRefMATH

30.

Liu W.J., Zhou S.L., Li H.L., Fang X.G.: Finite linear spaces admitting a Ree simple group. Eur. J. Comb. 25, 311–325 (2004).MathSciNetCrossRefMATH

31.

Liu W.J., Dai S.J., Gong L.Z.: Almost simple groups with socle \({}^3D_4(q)\) act on finite linear spaces. Sci. China Math. 49, 1768–1776 (2006).CrossRefMATH

32.

Liu W.J., Li S.Z., Gong L.Z.: Almost simple groups with socle \(\rm {Ree}(q)\) acting on finite linear spaces. Eur. J. Comb. 27, 788–800 (2006).MathSciNetCrossRefMATH

33.

Mann A., Tuan N.D.: Block-transitive point-imprimitive \(t\)-designs. Geom. Dedicata. 88, 81–90 (2001).MathSciNetCrossRefMATH

34.

Maróti A.: On the orders of primitive groups. J. Algebra 258, 631–640 (2002).MathSciNetCrossRefMATH

35.

Tang J.X., Liu W.J., Wang J.H.: Groups PSL \((n, q)\) and \(3\)-\((v, k, 1)\) designs. ARS Comb. 110, 217–226 (2013).MathSciNetMATH

36.

Tang J.X., Liu W.J., Chen J.: On block-transitive \(6\)-\((v, k,1)\)-designs with \(k\) at most \(10000\). Algebra Colloq. 21, 231–234 (2014).MathSciNetCrossRefMATH

37.

Yin F.G., Feng Y.Q., Xia B.: The smallest vertex-primitive \(2\)-arc-transitive digraph. J. Algebra 626 (2023). https://doi.org/10.1016/j.jalgebra.2023.02.028

38.

Zhou S.L.: Block primitive \(2\)-\((v, k,1)\) designs admitting a Ree simple group. Eur. J. Comb. 23, 1085–1090 (2002).MathSciNetCrossRefMATH

39.

Zhou S.L.: Block primitive \(2\)-\((v, k,1)\) designs admitting a Ree group of characteristic Two. Des. Codes Cryptogr. 36, 159–169 (2005).MathSciNetCrossRefMATH

40.

Zhou S.L., Li H.L., Liu W.J.: The Ree groups \({}^2G_2(q)\) and \(2\)-\((v, k,1)\) block designs. Discret. Math. 224, 251–258 (2000).CrossRefMATH

Title: Block-transitive 3-(v, k, 1) designs associated with alternating groups
Authors: Ting Lan
Weijun Liu
Fu-Gang Yin
Publication date: 15-05-2023
Publisher: Springer US
Published in: Designs, Codes and Cryptography / Issue 8/2023
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-023-01215-7

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Block-transitive 3-(v, k, 1) designs associated with alternating groups

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