Biplanes with flag-transitive automorphism groups of almost simple type, with classical socle
Eugenia O'Reilly-Regueiro
Universidad Nacional Autónoma de México Instituto de Matemáticas Mexico DF 04510 Mexico
DOI: 10.1007/s10801-007-0070-7
Abstract
In this paper we prove that if a biplane D admits a flag-transitive automorphism group G of almost simple type with classical socle, then D is either the unique (11,5,2) or the unique (7,4,2) biplane, and G\leq PSL 2(11) or PSL 2(7), respectively. Here if X is the socle of G (that is, the product of all its minimal normal subgroups), then X\? G\leq Aut\thinspace G and X is a simple classical group.
Pages: 529–552
Keywords: keywords automorphism group; biplanes; flag-transitive
Full Text: PDF
References
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2. Aschbacher, M. (1984). On the maximal subgroups of the finite classical groups. Invent. Math., 76, 469-514.
3. Assmus, E. F. Jr., Mezzaroba, J. A., & Salwach, C. J. (1977). Planes and biplanes. In Proceedings of the 1976 Berlin Combinatorics Conference. Vancerredle.
4. Assmus, E. F. Jr., & Salwach, C. J. (1979). The (16,6,2) designs. Int. J. Math. Math. Sci., 2(2), 261- 281.
5. Bannai, E., Hao, S., & Song, S.-Y. (1990). Character tables of the association schemes of finite orthogonal groups on the non-isotropic points. J. Comb. Theory Ser. A, 54, 164-200.
6. Cameron, P. J. (1973). Biplanes. Math. Z., 131, 85-101.
7. Colbourn, C. J., & Dinitz, J. H. (1996). The CRC handbook of combinatorial designs. Boca Raton: CRC.
8. Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., & Wilson, R. A. (1985). Atlas of finite groups. London: Oxford University Press.
9. Davies, H. (1987). Flag-transitivity and primitivity. Discret. Math., 63, 91-93.
10. Denniston, R. H. F. (1980). On biplanes with 56 points. Ars. Comb., 9, 167-179.
11. Hall, M. Jr., Lane, R., & Wales, D. (1970). Designs derived from permutation groups. J. Comb. Theory, 8, 12-22.
12. Huppert, B. (1967). Endliche Gruppen. Berlin: Springer.
13. Hussain, Q. M. (1945). On the totality of the solutions for the symmetrical incomplete block designs λ= 2, k = 5 or
6. Sankhya, 7, 204-208.
14. Kantor, W. (1985). Classification of 2-transitive symmetric designs. Graphs Comb., 1, 165-166.
15. Kleidman, P. B. (1987). The subgroup structure of some finite simple groups. PhD thesis, University of Cambridge.
16. Kleidman, P. B. (1987). The maximal subgroups of the finite 8-dimensional orthogonal groups + P (q) and of their automorphism groups. J. Algebra, 110, 172-242. 8
17. Kleidman, P. B., & Liebeck, M. W. (1990). The subgroup structure of the finite classical groups. London math. soc. lecture note series, Vol.
129. Cambridge: Cambridge Univ. Press.
18. Liebeck, M. W. (1985). On the orders of maximal subgroups of the finite classical groups. Proc. Lond. Math. Soc., 50, 426-446.
19. Liebeck, M. W. (1987). The affine permutation groups of rank
3. Proc. Lond. Math. Soc., 54, 477- 516.
20. Liebeck, M. W., & Saxl, J. (1987). On the orders of maximal subgroups of the finite exceptional groups of Lie type. Proc. Lond. Math. Soc., 55, 299-330.
21. Liebeck, M. W., Saxl, J., & Seitz, G. M. (1987). On the overgroups of irreducible subgroups of the finite classical groups. Proc. Lond. Math. Soc., 55, 507-537.
22. Liebeck, M. W., Praeger, C. E., & Saxl, J. (1988). On the O'Nan-Scott theorem for finite primitive permutation groups. J. Aust. Math. Soc. (Ser. A), 44, 389-396.
23. Liebeck, M. W., Praeger, C. E., & Saxl, J. (1988). On the 2-closures of finite permutation groups. J. Lond. Math. Soc., 37, 241-264.
24. O'Reilly Regueiro, E. (2005). On primitivity and reduction for flag-transitive symmetric designs. J. Comb. Theory Ser. A, 109, 135-148.
25. O'Reilly Regueiro, E. (2005). Biplanes with flag-transitive automorphism groups of almost simple type, with alternating or sporadic socle. Eur. J. Comb., 26, 577-584.
26. Salwach, C. J., & Mezzaroba, J. A. (1978). The four biplanes with k =
9. J. Comb. Theory Ser. A, 24, 141-145.
27. Saxl, J. (2002). On finite linear spaces with almost simple flag-transitive automorphism groups. J. Comb. Theory Ser. A, 100(2), 322-348.
28. Seitz, G. M. (1973). Flag-transitive subgroups of Chevalley groups. Ann. Math., 97(1), 27-56.