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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.

20J06 Cohomology of groups
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
Full Text: DOI
[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
[2] Hilton, P.J.; Stammbach, U., A course in homological algebra, () · Zbl 0238.18006
[3] Horadam, K.J.; de Launey, W., Cocyclic development of designs, J. algebraic combin., 2, 267-290, (1993) · Zbl 0785.05019
[4] Johnson, D.L., Presentation of groups, () · Zbl 0696.20027
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