Calculation of cocyclic matrices
type: Item
instance of: scholarly article
title: Calculation of cocyclic matrices
zbMATH Open document ID: 0867.20043
DOI: 10.1016/0022-4049(95)00135-2
published in: Journal of Pure and Applied Algebra
publication date: 1997-06-11T00:00:00Z
review text: 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 \textit{K. J. Horadam} and \textit{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.
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 05B20
zbMATH DE Number: 933077
zbMATH Keywords: finite groups
zbMATH Keywords: cohomology groups
zbMATH Keywords: \(2\)-cocycles
zbMATH Keywords: cocyclic matrices
zbMATH Keywords: Hadamard matrices
author: D. L. Flannery
reviewed by: Valeriy B. Mnukhin
MaRDI profile type: Publication
full work available at URL: 0022-4049(95)00135-2
OpenAlex ID: W2046506885
cites work: A weak difference set construction for higher dimensional designs
cites work: Q5648522
cites work: Cocyclic development of designs
cites work: Q4040881
cites work: On linearizable irreducible projective representations of finite groups
cites work: Q3755633
cites work: Q5343374
description: scientific article; zbMATH DE number 933077