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Elementary Abelian Cartesian Groups Cartesian Groups

Published online by Cambridge University Press:  20 November 2018

John L. Hayden
John L. Hayden*
Affiliation:
Bowling Green State University, Bowling Green, Ohio
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Throughout the paper, G will denote an additively written, but not always abelian, group of finite order n; and X = (xij) will denote a square matrix of order n with entries from G and whose rows and columns are numbered 0, 1, …, n − 1. We call X a cartesian array (afforded by G) if

(1.1) The sequence {−xmi + xki, i = 0,…, n – 1} contains all elements of G whenever k ≠ m.

By a theorem of Jungnickel (see Theorem 2.2 in [5]), the transpose of a cartesian array is also a cartesian array. We call G a cartesian group if there is a cartesian array X afforded by G. In this case, we also call (G, X) a cartesian pair.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 6 , 01 December 1988 , pp. 1315 - 1321
Copyright
Copyright © Canadian Mathematical Society 1988

References

1. Bruck, R. H., Finite nets II, uniqueness and imbedding, Pacific J. Math. 13 (1963), 421–457.Google Scholar
2. Drake, D. A., Partial λ-geometries and generalized Hadamard matrices over groups, Can. J. Math. 31 (1979), 617–627.Google Scholar
3. Hayden, J. L., A representation theory for Cartesian groups, Algebras, Groups and Geometries 2 (1985), 399–427.Google Scholar
4. Hughes, D. R. and Piper, F. C., Projective planes (Springer-Verlag, 1973).Google Scholar
5. Jungnickel, D., On difference matrices, resolvable transversal designs and generalized Hadamard matrices, Math. Zeitschrift 167 (1979), 49–60.Google Scholar
6. Østrom, >T. G., Finite planes with a single (p, L) transitivity, Arch. Math. 15 (1964), 378–384.Google Scholar
7. Østrom, T. G., Semi-translation planes, Trans. Amer. Math. Soc. 111 (1964), 1–18.Google Scholar