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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 [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 [5] Karpilovsky, G., Projective representations of finite groups, () · Zbl 0316.20007 [6] Karpilovsky, G., The Schur multiplier, () · Zbl 0619.20001 [7] MacLane, S., Homology, (1963), Springer Berlin · Zbl 0133.26502
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