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The Inverse Multiplier for Abelian Group Difference Sets

Published online by Cambridge University Press:  20 November 2018

E. C. Johnsen
E. C. Johnsen*
Affiliation:
National Bureau of Standards, Washington, D.C.
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In (1) Bruck introduced the notion of a difference set in a finite group. Let G be a finite group of v elements and let D = {di}, i = 1, . . . , k be a k-subset of G such that in the set of differences {di-1dj} each element ≠ 1 in G appears exactly λ times, where 0 < λ < k <v— 1. When this occurs we say that (G, D) is a v, k, λ group difference set. Bruck showed that this situation is equivalent to the one where the differences {didj-1} are considered instead, and that a v, k, λ group difference set is equivalent to a transitive v, k, λ configuration, i.e., a v, k, λ configuration which has a collineation group which is transitive and regular on the elements (points) and on the blocks (lines) of the configuration. Among the parameters v, k and λ, then, we have the relation shown by Ryser (5)

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 16 , 1964 , pp. 787 - 796
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Bruck, R. H., Difference sets in a finite group, Trans. Amer. Math. Soc, 78 (1955), 464–481.Google Scholar
2. Chowla, S. and Ryser, H. J., Combinatorial problems, Can. J. Math., 2 (1950), 93–99.Google Scholar
3. Mann, H. B., Balanced incomplete block designs and abelian difference sets, Illinois J. Math., 8 (1964), 252–261.Google Scholar
4. Menon, P., Difference sets in abelian groups, Proc. Amer. Math. Soc, 11 (1960), 368–376.Google Scholar
5. Ryser, H. J., A note on a combinatorial problem, Proc. Amer. Math. Soc, 1 (1950), 422–424.Google Scholar
6. Ryser, H. J., Combinatorial mathematics, Carus Maht. Monograph. No. 14 (Math. Ass'n. Amer., 1963).Google Scholar