×

Cocyclic Hadamard matrices and Hadamard groups are equivalent. (English) Zbl 0889.05032

It is proved that the existence of cocyclic Hadamard matrices is equivalent to the existence of certain Hadamard groups. A procedure for the construction of Hadamard groups is described.

MSC:

05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
20E22 Extensions, wreath products, and other compositions of groups
05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
20J05 Homological methods in group theory
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Baliga, A.; Horadam, K. J., Cocyclic Hadamard matrices over \(Z_t\)×\(Z_2^2\), Australas. J. Combin., 11, 123-134 (1995) · Zbl 0838.05017
[2] Colbourn, C. J.; Dinitz, J. H., The CRC Handbook of Combinatorial Designs (1996), CRC Press: CRC Press Boca Raton · Zbl 0836.00010
[3] de Launey, W.; Horadam, K. J., A weak difference set construction for higher dimensional designs, Des. Codes Cryptogr., 3, 75-87 (1993) · Zbl 0838.05019
[4] Flannery, D. L., Calculation of cocyclic matrices, J. Pure Appl. Algebra, 112, 181-190 (1996) · Zbl 0867.20043
[5] Flannery, D. L., The finite irreducible linear 2-groups of degree 4, Mem. Amer. Math. Soc., 129 (1997) · Zbl 0927.20026
[6] Geramita, A. V.; Seberry, J., Orthogonal Designs: Quadratic Forms and Hadamard matrices (1979), Dekker: Dekker New York · Zbl 0411.05023
[7] Horadam, K. J.; de Launey, W., Cocyclic development of designs, J. Algebraic Combin., 2, 267-290 (1993) · Zbl 0785.05019
[8] Horadam, K. J., Progress in cocyclic matrices, Congr. Numer., 118, 161-171 (1996) · Zbl 0898.05011
[9] K. J. Horadam, D. L. Flannery, W. de Launey, 1997, Cocyclic Hadamard matrices and difference sets; K. J. Horadam, D. L. Flannery, W. de Launey, 1997, Cocyclic Hadamard matrices and difference sets · Zbl 0956.05026
[10] Ito, N., On Hadamard groups, J. Algebra, 168, 981-987 (1994) · Zbl 0906.05012
[11] Ito, N., On Hadamard groups, II, J. Algebra, 169, 936-942 (1994) · Zbl 0808.05016
[12] Ito, N., On Hadamard groups, III, Kyushu J. Math. (1996)
[13] Ito, N., Remarks on Hadamard groups, Kyushu J. Math., 50, 1-9 (1996)
[14] Ito, N.; Okamoto, T., On Hadamard groups of order 72, Algebra Colloq., 3, 307-324 (1996) · Zbl 0869.05017
[15] Karpilovsky, G., The Schur Multiplier. The Schur Multiplier, London Math. Soc. Monographs, New Ser., 2 (1987), Clarendon: Clarendon Oxford · Zbl 0619.20001
[16] Mac Lane, S., Homology (1963), Springer-Verlag: Springer-Verlag Berlin/Göttingen/Heidelberg
[17] W. Nickel, 1993, Central Extensions of Polycyclic Groups, Australian National University; W. Nickel, 1993, Central Extensions of Polycyclic Groups, Australian National University
[18] Robinson, D. J., Applications of cohomology to the theory of groups, (Campbell, C. M.; Robertson, E. F., Groups—St Andrews 1981 (1982), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 46-80
[19] Scott, W. R., Group Theory (1987), Dover: Dover New York
[20] Wallis, W. D., Combinatorial Designs (1988), Dekker: Dekker New York · Zbl 0637.05004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.