Biplanes with flag-transitive automorphism groups of almost simple type, with exceptional socle of Lie type
Eugenia O'Reilly-Regueiro
Universidad Nacional Autónoma de México Instituto de Matemáticas Circuito Exterior, Ciudad Universitaria México DF 04510 Mexico
DOI: 10.1007/s10801-007-0098-8
Abstract
In this paper we prove that there is no biplane admitting a flag-transitive automorphism group of almost simple type, with exceptional socle of Lie type. A biplane is a ( v, k,2)-symmetric design, and a flag is an incident point-block pair. A group G is almost simple with socle X if X is the product of all the minimal normal subgroups of G, and X\? G\leq Aut ( G).
Throughout this work we use the classification of finite simple groups, as well as results from P.B. Kleidman's Ph.D. thesis which have not been published elsewhere.
Pages: 479–491
Keywords: keywords automorphism group; flag-transitive; primitive group; symmetric design
Full Text: PDF
References
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2. Aschbacher, M.: On the maximal subgroups of the finite classical groups. Invent. Math. 76, 469-514 (1984)
3. Assmus, E.F. Jr., Mezzaroba, J.A., Salwach, C.J.: Planes and biplanes. In: Proceedings of the 1976 Berlin Combinatorics Conference, Vancerredle (1977)
4. Assmus, E.F. Jr., Salwach, C.J.: The (16,6,2) designs. Int. J. Math. Math. Sci. 2(2), 261-281 (1979)
5. Cameron, P.J.: Biplanes. Math. Z. 131, 85-101 (1973)
6. Cohen, A.M., Liebeck, M.W., Saxl, J., Seitz, G.M.: The local maximal subgroups of exceptional groups of Lie type, finite and algebraic. Proc. Lond. Math. Soc. (3) 64, 21-48 (1992)
7. Colburn, C.J., Dinitz, J.H.: The CRC Handbook of Combinatorial Designs. CRC Press, Boca Raton (1996)
8. Cooperstein, B.N.: Minimal degree for a permutation representation of a classical group. Isr. J. Math. 30, 213-235 (1978)
9. Denniston, R.H.F.: On biplanes with 56 points. Ars. Comb. 9, 167-179 (1980)
10. Hall, M. Jr., Lane, R., Wales, D.: Designs derived from permutation groups. J. Comb. Theory 8, 12-22 (1970)
11. Hussain, Q.M.: On the totality of the solutions for the symmetrical incomplete block designs λ= 2, k = 5 or
6. Sankhya 7, 204-208 (1945)
12. Kleidman, P.B.: The subgroup structure of some finite simple groups. PhD thesis, University of Cambridge (1987)
13. Kleidman, P.B.: The maximal subgroups of the Chevalley groups G2(q) with q odd, the Ree groups 2G2(q), and their automorphism groups. J. Algebra 117, 30-71 (1998)
14. Kleidman, P.B., Liebeck, M.W.: The Subgroup Structure of the Finite Classical Groups. London Mathematical Society Lecture Note Series, vol.
129. Cambridge University Press, Cambridge (1990)
15. Liebeck, M.W.: On the orders of maximal subgroups of the finite classical groups. Proc. Lond. Math. Soc. 50, 426-446 (1985)
16. Liebeck, M.W., Praeger, C.E., Saxl, J.: The maximal factorizations of the finite simple groups and their automorphism groups. Mem. Am. Math. Soc. 86(432), 1-151 (1990)
17. Liebeck, M.W., Praeger, C.E., Saxl, J.: On the O'Nan-Scott theorem for finite primitive permutation groups. J. Austral. Math. Soc. (Ser. A) 44, 389-396 (1988)
18. Liebeck, M.W., Saxl, J.: The primitive permutation groups of odd degree. J. Lond. Math. Soc. 31, 250-264 (1985)
19. Liebeck, M.W., Saxl, J.: The finite primitive permutation groups of rank three. Bull. Lond. Math. Soc. 18, 165-172 (1986)
20. Liebeck, M.W., Saxl, J.: On the orders of maximal subgroups of the finite exceptional groups of Lie type. Proc. Lond. Math. Soc. 55, 299-330 (1987)
21. Liebeck, M.W., Saxl, J., Seitz, G.M.: Subgroups of maximal rank in finite exceptional groups of Lie type. Proc. Lond. Math. Soc. 65, 297-325 (1992)
22. Liebeck, M.W., Saxl, J., Seitz, G.M.: On the overgroups of irreducible subgroups of the finite classical groups. Proc. Lond. Math. Soc. 55, 507-537 (1987)
23. Liebeck, M.W., Saxl, J., Testerman, D.M.: Simple subgroups of large rank in groups of Lie type. Proc. Lond. Math. Soc. (3) 72, 425-457 (1996)
24. Liebeck, M.W., Seitz, G.M.: Maximal subgroups of exceptional groups of Lie type, finite and algebraic. Geom. Dedic. 35, 353-387 (1990)
25. Liebeck, M.W., Seitz, G.M.: On finite subgroups of exceptional algebraic groups. J. Reine Angew. Math. 515, 25-72 (1999)
26. Liebeck, M.W., Shalev, A.: The probability of generating a finite simple group. Geom. Dedic. 56, 103-113 (1995)
27. Malle, G.: The maximal subgroups of 2F4(q2). J. Algebra 139, 53-69 (1991)
28. O'Reilly Regueiro, E.: On primitivity and reduction for flag-transitive symmetric designs. J. Comb. Theory Ser. A 109, 135-148 (2005)
29. O'Reilly Regueiro, E.: Biplanes with flag-transitive automorphism groups of almost simple type, with alternating or sporadic socle. Eur. J. Comb. 26, 577-584 (2005)
30. O'Reilly-Regueiro, E.: Biplanes with flag-transitive automorphism groups of almost simple type, with classical socle. J. Algebr. Comb. (2007, to appear)
31. Salwach, C.J., Mezzaroba, J.A.: The four biplanes with k =
9. J. Comb. Theory Ser. A 24, 141-145 (1978)
32. Saxl, J.: On finite linear spaces with almost simple flag-transitive automorphism groups. J. Comb.
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