A census of highly symmetric combinatorial designs
Michael Huber
Mathematisches Institut der Universität Tübingen Auf der Morgenstelle 10 72076 Tübingen Germany
DOI: 10.1007/s10801-007-0065-4
Abstract
As a consequence of the classification of the finite simple groups, it has been possible in recent years to characterize Steiner t-designs, that is t-( v, k,1) designs, mainly for t=2, admitting groups of automorphisms with sufficiently strong symmetry properties. However, despite the finite simple group classification, for Steiner t-designs with t>2 most of these characterizations have remained long-standing challenging problems. Especially, the determination of all flag-transitive Steiner t-designs with 3\leq t\leq 6 is of particular interest and has been open for about 40 years (cf. Delandtsheer (Geom. Dedicata 41, p. 147, 1992 and Handbook of Incidence Geometry, Elsevier Science, Amsterdam, 1995, p. 273), but presumably dating back to 1965).
The present paper continues the author's work (see Huber (J. Comb. Theory Ser. A 94, 180-190, 2001; Adv. Geom. 5, 195-221, 2005; J. Algebr. Comb., 2007, to appear)) of classifying all flag-transitive Steiner 3-designs and 4-designs. We give a complete classification of all flag-transitive Steiner 5-designs and prove furthermore that there are no non-trivial flag-transitive Steiner 6-designs. Both results rely on the classification of the finite 3-homogeneous permutation groups. Moreover, we survey some of the most general results on highly symmetric Steiner t-designs.
Pages: 453–476
Keywords: keywords Steiner designs; flag-transitive group of automorphisms; 3-homogeneous permutation groups
Full Text: PDF
References
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34. Liebeck, M. W. (1998). The classification of finite linear spaces with flag-transitive automorphism groups of affine type. J. Comb. Theory Ser. A, 84, 196-235.
35. Livingstone, D., & Wagner, A. (1965). Transitivity of finite permutation groups on unordered sets. Math. Z., 90, 393-403.
36. Lüneburg, H. (1965). Fahnenhomogene quadrupelsysteme. Math. Z., 89, 82-90.
37. Ray-Chaudhuri, D. K., & Wilson, R. M. (1975). On t -designs. Osaka J. Math., 12, 737-744.
38. Saxl, J. (2002). On finite linear spaces with almost simple flag-transitive automorphism groups. J. Comb. Theory Ser. A, 100, 322-348.
39. Tits, J. (1964). Sur les systèmes de Steiner associés aux trois “grands” groupes de Mathieu. Rend. Math., 23, 166-184.
40. Witt, E. (1938). Über Steinersche Systeme. Abh. Math. Sem. Univ. Hamburg, 12, 265-275.
2. Beth, Th., Jungnickel, D., & Lenz, H. (1999). Design theory, vols. I and II. Encyclopedia of math. and its applications (Vols. 69/78). Cambridge: Cambridge Univ. Press.
3. Block, R. E. (1965). Transitive groups of collineations on certain designs. Pac. J. Math., 15, 13-18.
4. Buekenhout, F. (1968). Remarques sur l'homogénéité des espaces linéaires et des systèmes de blocs. Math. Z., 104, 144-146.
5. Buekenhout, F., Delandtsheer, A., & Doyen, J. (1988). Finite linear spaces with flag-transitive groups. J. Comb. Theory Ser. A, 49, 268-293.
6. Buekenhout, F., Delandtsheer, A., Doyen, J., Kleidman, P. B., Liebeck, M. W., & Saxl, J. (1990). Linear spaces with flag-transitive automorphism groups. Geom. Dedicata, 36, 89-94.
7. Cameron, P. J. (1976). Parallelisms of complete designs. London math. soc. lecture note series (Vol. 23). Cambridge: Cambridge Univ. Press. J Algebr Comb (2007) 26: 453-476
8. Cameron, P. J. (1981). Finite permutation groups and finite simple groups. Bull. Lond. Math. Soc., 13, 1-22.
9. Cameron, P. J., Praeger, C. E. (1993). Block-transitive t -designs, II: large t . In F. De Clerck, et al. (Eds.), Finite geometry and combinatorics, deinze,
1992. London math. soc. lecture note series (Vol. 191, pp. 103-119). Cambridge: Cambridge Univ. Press.
10. Carmichael, R. D. (1937). Introduction to the theory of groups of finite order. Boston: Ginn; Reprint: New York: Dover Publications (1956).
11. Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., & Wilson, R. A. (1985). Atlas of finite groups. Oxford: Clarendon Press.
12. Delandtsheer, A. (1992). Finite (line, plane)-flag-transitive planar spaces. Geom. Dedicata, 41, 145- 153.
13. Delandtsheer, A. (1995). Dimensional linear spaces. In F. Buekenhout (Ed.), Handbook of incidence geometry (pp. 193-294). Amsterdam: Elsevier Science.
14. Delandtsheer, A. (2001). Finite flag-transitive linear spaces with alternating socle. In A. Betten, et al. (Eds.), Algebraic combinatorics and applications. Proc. Euroconf., Gößweinstein, 1999 (pp. 79-88). Berlin: Springer.
15. Dembowski, P. (1968). Finite geometries. Berlin: Springer; Reprint: Springer (1997).
16. Dickson, L. E. (1901). Linear groups with an exposition of the Galois field theory. Leipzig: Teubner; Reprint: New York: Dover Publications (1958).
17. Foulser, D. A. (1964). The flag-transitive collineation groups of the finite Desarguesian affine planes. Can. J. Math., 16, 443-472.
18. Foulser, D. A. (1964). Solvable flag-transitive affine groups. Math. Z., 86, 191-204.
19. Gorenstein, D. (1982). Finite simple groups. An introduction to their classification. New York: Plenum.
20. Higman, D. G., & McLaughlin, J. E. (1961). Geometric ABA-groups. Ill. J. Math., 5, 382-397.
21. Huber, M. (2001). Classification of flag-transitive Steiner quadruple systems. J. Comb. Theory Ser. A, 94, 180-190.
22. Huber, M. (2005). The classification of flag-transitive Steiner 3-designs. Adv. Geom., 5, 195-221.
23. Huber, M. (2005). On highly symmetric combinatorial designs. Tübingen: Habilitationsschrift. Univ. Tübingen; Aachen: Shaker (2006).
24. Huber, M. The classification of flag-transitive Steiner 4-designs, J. Algebr. Comb. (2007, to appear), 25. .
25. Huppert, B. (1967). Endliche gruppen I. Berlin: Springer.
26. Kantor, W. M. (1972). k-homogeneous groups. Math. Z., 124, 261-265.
27. Kantor, W. M. (1985). Homogeneous designs and geometric lattices. J. Comb. Theory Ser. A, 38, 66-74.
28. Kantor, W. M. (1985). Flag-transitive planes. In C. A. Baker, L. M. Batten (Eds.), Finite geometries, Winnipeg,
1984. Lecture notes in pure and applied math. (Vol. 103, pp. 179-181). New York: Dekker.
29. Kantor, W. M. (1987). Primitive permutation groups of odd degree, and an application to finite projective planes. J. Algebra, 106, 15-45.
30. Kantor, W. M. (1993). 2-transitive and flag-transitive designs. In D. Jungnickel, et al. (Eds.), Coding theory, design theory, group theory (pp. 13-30). Proc. Marshall Hall Conf., Burlington,
1990. New York: Wiley.
31. Kleidman, P. B. (1990). The finite flag-transitive linear spaces with an exceptional automorphism group. In E. S. Kramer, S. S. Magliveras (Eds.), Finite geometries and combinatorial designs, Lincoln,
1987. Contemp. math. (Vol. 111, pp. 117-136). Providence: Am. Math. Soc..
32. 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.
33. Liebeck, M. W. (1987). The affine permutation groups of rank three. Proc. Lond. Math. Soc. (3), 54, 477-516.
34. Liebeck, M. W. (1998). The classification of finite linear spaces with flag-transitive automorphism groups of affine type. J. Comb. Theory Ser. A, 84, 196-235.
35. Livingstone, D., & Wagner, A. (1965). Transitivity of finite permutation groups on unordered sets. Math. Z., 90, 393-403.
36. Lüneburg, H. (1965). Fahnenhomogene quadrupelsysteme. Math. Z., 89, 82-90.
37. Ray-Chaudhuri, D. K., & Wilson, R. M. (1975). On t -designs. Osaka J. Math., 12, 737-744.
38. Saxl, J. (2002). On finite linear spaces with almost simple flag-transitive automorphism groups. J. Comb. Theory Ser. A, 100, 322-348.
39. Tits, J. (1964). Sur les systèmes de Steiner associés aux trois “grands” groupes de Mathieu. Rend. Math., 23, 166-184.
40. Witt, E. (1938). Über Steinersche Systeme. Abh. Math. Sem. Univ. Hamburg, 12, 265-275.
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