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Calculation of cocyclic matrices. (English) Zbl 0867.20043
Let \(G\) be a finite group, \(U\) be a \(G\)-module and \(H^2(G,U)\) the second cohomology group of \(G\) with coefficients in \(U\). Note that a 2-cocycle \(\psi\) is naturally displayed as a cocyclic matrix whose rows and columns are indexed by the elements of \(G\) and whose entry in the position \((g,h)\) is \(\psi(g,h)\). The cocyclic matrices with coefficients in \(\mathbb{Z}_2\) are closely related to Hadamard matrices and may consequently provide a new way of generating designs, see K. J. Horadam and W. de Launey [J. Algebr. Comb. 2, No. 3, 267-290 (1993; Zbl 0785.05019)].
In this paper the author provides a method of explicitly determining cocyclic matrices of representatives for all 2-cocycle classes in \(H^2(G,U)\), when \(U\) is a finitely generated \(G\)-module trivial under the action of \(G\). The method is based on the Universal Coefficient Theorem. Also symmetry properties of cocyclic matrices are investigated.

MSC:
20J06 Cohomology of groups
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
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References:
[1] de Launey, W.; Horadam, K.J., A weak difference set construction for higher dimensional designs, Designs codes cryptography, 3, 75-87, (1993) · Zbl 0838.05019
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