Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-02-12T08:40:54.176Z Has data issue: false hasContentIssue false
  • English
  • Français

On Automorphism Groups of Divisible Designs

Published online by Cambridge University Press:  20 November 2018

Dieter Jungnickel
Dieter Jungnickel*
Affiliation:
Justus-Liebig-Universität Giessen, Giessen, F.R. Germany
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A (group) divisible design is a tactical configuration for which the v points are split into m classes of n each, such that points have joining number λ (resp. λ2) if and only if they are in the same (resp. in different) classes. We are interested in such designs with a nice automorphism group. We first investigate divisible designs with equally many points and blocks admitting an automorphism group acting regularly on all points and on all blocks, i.e., with a Singer group (Singer [50] obtained the first result in this direction for the finite projective spaces).

As in the case of block designs, one may expect a divisible design with a Singer group to be equivalent to some sort of difference set; as it turns out, one here obtains a generalisation of the relative difference sets of Butson and Elliott [11] and [20].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 2 , 01 April 1982 , pp. 257 - 297
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. Baumert, L. D., Cyclic difference sets, Springer Lecture Notes 182 (Berlin-Heidelberg- New York, 1971).Google Scholar
2. Beker, H. and Piper, F. C., Some designs which admit strong tactical decompositions, J. Comb. Th. A 22 (1977), 38–42.Google Scholar
3. Berman, G., Families of generalized weighing matrices, Can. J. Math. 30 (1978), 1016–1028.Google Scholar
4. Bose, R. C., An affine analogue of Singer's theorem, J. Ind. Math. Soc. 6 (1942), 1–15.Google Scholar
5. Bose, R. C., Symmetric group divisible designs with the dual property, J. Stat. Planning Inf. 1 (1977), 87–101.Google Scholar
6. Bose, R. C. and Connor, W. S., Combinatorial properties of group divisible incomplete block designs, Ann. Math. Stat. 28 (1952), 367–383.Google Scholar
7. Bose, R. C., Shrikhande, S. S. and Bhattacharya, K. N., On the construction of group divisible incomplete block designs, Ann. Math. Stat. 2J+ (1953), 167–195.Google Scholar
8. Bose, R. C., Shrikhande, S. S. and Parker, E. T., Further results on the construction of mutually orthogonal Latin squares and the falsity of Ruler's conjecture, Can. J. Math. 12 (1960), 189–203.Google Scholar
9. Bruck, R. H., Difference sets in a finite group, Trans. Amer. Math. Soc. 78 (1955), 464–481.Google Scholar
10. Butson, A. T., Generalized Hadamard matrices, Proc. Amer. Math. Soc. 13 (1962), 894–898.Google Scholar
11. Butson, A. T., Relations among generalized Hadamard matrices, relative difference sets and maximal length linear recurring sequences, Can. J. Math. 15 (1963), 42–48.Google Scholar
12. Cameron, P. J., Delsarte, P. and Geoethals, J. M., Hemisystems, orthogonal configurations and dissipative conference matrices, Philips J. Res. 34 (1979), 147–162.Google Scholar
13. Connor, W. S., Some relations among the blocks of symmetrical group divisible designs, Ann. Math. Stat. 23 (1952), 602–609.Google Scholar
14. Delsarte, P., Orthogonal matrices over a group and related tactical configurations, M.B.L.E. Report R90 (1968).Google Scholar
15. Delsarte, P., Goethals, J. M. and Seidel, J. J., Orthogonal matrices with zero diagonal II, Can. J. Math. 23 (1971), 816–832.Google Scholar
16. Dembowski, P., Finite geometries (Springer, Berlin-Heidelberg-New York, 1968).Google Scholar
17. Drake, D. A., Partial X-geometries and generalized Hadamard matrices over groups, Can. J. Math. 31 (1979), 617–627.Google Scholar
18. Drake, D. A. and Jungnickel, D., Klingenberg structures and partial designs I. Congruence relations and solutions, J. Stat. Planning Inf. 1 (1977), 265–287.Google Scholar
19. Drake, D. A. and Jungnickel, D., Klingenberg structures and partial designs II: Regularity and uniformity, Pac. J. Math. 77 (1978), 389–415.Google Scholar
20. J. E. H., Elliott and Butson, A. T., Relative difference sets, Illinois J. Math. 10 (1966), 517–531.Google Scholar
21. Goethals, J. M. and Seidel, J. J., Orthogonal matrices with zero diagonal, Can. J. Math. 19 (1967), 1001–1010.Google Scholar
22. Hall, M. Jr., Combinatorial theory (Blaisdell, Waltham, 1967).Google Scholar
23. Hoffman, A. J., Cyclic affine planes, Can. J. Math. 4 (1952), 295–301.Google Scholar
24. Hughes, D. R., Partial difference sets, Amer. J. Math. 78 (1956), 650–674.Google Scholar
25. Hughes, D. R., Biplanes and semibiplanes, In: Combinatorial mathematics, Springer Lecture Notes No. 686, 55-58 (Berlin-Heidelberg-New York, 1978).Google Scholar
26. Hughes, D. R., On designs. To appear in Geometries and groups, Springer Lecture Notes 893.Google Scholar
27. Hughes, D. R. and Piper, F. C., Projective planes (Springer, Berlin-Heidelberg-New York, 1973).Google Scholar
28. Johnson, D. M., Dulmage, A. K. and Mendelsohn, N. S., Or thorn or phis m s of groups and orthogonal Latin squares, Can. J. Math. 13 (1961), 356–372.Google Scholar
29. Jungnickel, D., On difference matrices and regular Latin squares, Abh. Math. Sem. Hamburg 50 (1980), 219–231.Google Scholar
30. Jungnickel, D., On difference matrices, resolvable transversal designs and generalized Hadamard matrices, Math. Z. 167 (1979), 49–60.Google Scholar
31. Jungnickel, D., Regular Hjelmslev planes, J. Comb. Th. A 26 (1979), 20–37.Google Scholar
32. Jungnickel, D., On an assertion of Dembowski, J. Geom. 12 (1979), 168–174.Google Scholar
33. Jungnickel, D., Some new combinatorial results on finite Klingenberg structures, Util. Math. 16 (1979), 249–269.Google Scholar
34. Jungnickel, D., On balanced regular Hjelmslev planes, Geom. Ded. 8 (1979), 445–462.Google Scholar
35. Jungnickel, D., A note on square divisible designs, J. Geom. 15 (1980), 153–157.Google Scholar
36. Ko, H.-P. and D. K., Ray-Chaudhuri, Multiplier theorems, J. Comb. Th. A 30 (1981), 134–157.Google Scholar
37. Mathon, R., Symmetric conference matrices of order pq2 + 1, Can. J. Math. 30 (1978), 321–333.Google Scholar
38. Mavron, V. C., Resolvable designs, affine designs and hypernets, Preprint 99, Freie Universitât Berlin (1979).Google Scholar
39. McFarland, R. L., A family of difference sets in non-cyclic groups, J. Comb. Th. A 15 (1973), 1–10.Google Scholar
40. Menon, P. K., On difference sets whose parameters satisfy a certain relation, Proc. Amer. Math. Soc. 13 (1962), 739–745.Google Scholar
41. Mielants, W., On the non-existence of a class of symmetric group divisible partial designs, J. Geom. 12 (1979), 89–98.Google Scholar
42. Mills, W. H., Some mutually orthogonal Latin squares, Proc. 8th Southeastern Conf. Comb., Graph Th. and Comp. (1977), 473–487.Google Scholar
43. Raghavarao, D., Constructions and combinatorial problems in design of experiments (Wiley, New York, 1971).Google Scholar
44. Rajkundlia, D., Some techniques for constructing new infinite families of balanced incomplete block designs, Ph.D. dissertation, Queen's University, Kingston (1978).Google Scholar
45. Seberry, J., A class of group divisible designs, Ars Combinatoria 6 (1978), 151–152.Google Scholar
46. Seberry, J., Some remarks on generalized Hadamard matrices and theorems of Rajkundlia on SBIBDs. In: Combinatorial mathematics VI, Springer Lecture Notes 748 (Berlin-Heidelberg-New York, 1979), 154–164.Google Scholar
47. Seberry, J., A construction for generalized Hadamard matrices, J. Stat. PI. Inf. 4 (1980), 365–368.Google Scholar
48. Shrikhande, S. S., Cyclic solutions of symmetrical group divisible designs, Calcutta Stat. Ass. Bull. 5 (1953), 36–39.Google Scholar
49. Shrikhande, S. S., Generalized Hadamard matrices and orthogonal arrays of strength two, Can. J. Math. 16 (1964), 736–740.Google Scholar
50. Singer, J., A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc. 43 (1938), 377–385.Google Scholar
51. Swart, H. and Vedder, K., Subplane replacement in projective planes, Aqua. Math. (to appear).Google Scholar
52. Wallis, W. D., Street, A. P. and Wallis, J. S., Combinatorics: Room squares, sum-free sets, Hadamard matrices, Springer Lecture Notes 292 (Berlin-Heidelberg-New York, 1972).Google Scholar
53. Wild, P. R., On semibiplanes, Ph.D. dissertation, University of London (1980).Google Scholar
54. Wilson, R. M., A few more squares, Proc. 5th Southeastern Conf. Comb., Graph Th. and Comp. (1974), 657–680.Google Scholar