Abstract
In this paper we complete a classification of finite linear spaces \({\mathcal{S}}\) with line size at most 12 admitting a line-transitive point-imprimitive subgroup of automorphisms. The examples are the Desarguesian projective planes of orders 4, 7, 9 and 11, two designs on 91 points with line size 6, and 467 designs on 729 points with line size 8.
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References
Betten A, Praeger CE. Searching for line-transitive point-imprimitive linear spaces with a grid structure, in preparation
Betten A, Cresp G, Praeger CE. Line-transitive point-imprimitive linear spaces: the grid case, submitted
Betten A, Delandtsheer A, Law M, Niemeyer AC, Praeger CE, Zhou S Linear spaces with a line-transitive point-imprimitive automorphism group and Fang-Li parameter gcd(k,r) at most eight, submitted for publication. (available at http://arxiv.org/abs/math.CO/0701629)
Camina AR, Mischke S (1996) Line-transitive automorphism groups of linear spaces. Electron J Combin 3(1): Research Paper 3, 16 pp. (electronic)
Camina AR and Praeger CE (1993). Line-transitive automorphism groups of linear spaces. Bull London Math Soc 25(4): 309–315
Camina AR and Siemons J (1989). Block transitive automorphism groups of 2 − (υ,k,1) block designs. J Combin Theory Ser A 51: 268–276
Conway JH, Curtis RT, Norton SP, Parker RA, Wilson RA (1985) Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray. Oxford University Press, Eynsham
Cresp G (2001) Searching for line-transitive, point-imprimitive, linear spaces. Honours Dissertation, The University of Western Australia, Perth. (available at http://arxiv.org/abs/math.CO/0604532)
Delandtsheer A and Doyen J (1989). Most block-transitive t-designs are point-primitive. Geom Dedicata 29(3): 307–310
Dixon JD and Mortimer B (1996). Permutation groups. Springer-Verlag, New York
The GAP Group (2004) GAP – Groups, Algorithms, and Programming, Version 4.4; (http://www.gap-system.org)
Lam CWH, Thiel L and Swiercz S (1989). The nonexistence of finite projective planes of order 10. Canad J Math 41(6): 1117–1123
Nickel W, Niemeyer AC, O’Keefe CM, Penttila T and Praeger CE (1992). The block-transitive, point-imprimitive 2-(729,8,1) designs. Appl Algebra Engrg Comm Comput 3: 47–61
O’Keefe CM, Penttila T and Praeger CE (1993). Block-transitive, point-imprimitive designs with λ = 1. Disc Math 115: 231–244
Praeger CE and Tuan ND (2003). Inequalities involving the Delandtsheer-Doyen parameters for finite line-transitive linear spaces. J Combin Theory Ser A 102(1): 268–276
Stoichev SD and Tonchev VD (1988). The automorphism groups of the known 2-(91,6,1) designs. C. R. Acad Bulgare Sci 41(4): 15–16
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Praeger, C.E., Zhou, S. Classification of line-transitive point-imprimitive linear spaces with line size at most 12. Des. Codes Cryptogr. 47, 99–111 (2008). https://doi.org/10.1007/s10623-007-9077-2
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DOI: https://doi.org/10.1007/s10623-007-9077-2