The Schur multiplier, fields, roots of unity, and a natural splitting
type: Item
instance of: scholarly article
title: The Schur multiplier, fields, roots of unity, and a natural splitting
zbMATH Open document ID: 0691.20010
DOI: 10.1016/0021-8693(90)90231-C
author: William F. Reynolds
published in: Journal of Algebra
publication date: 1990-01-01T00:00:00Z
review text: Let G be a finite group and let \(\Omega\) be a subgroup of \({\mathbb{C}}^*\) which contains a root of unity of order \(| G|\). The universal coefficient theorem yields a split exact sequence \[ 1\quad \to \quad Ext(G/G',\Omega)\quad \to \quad H^ 2(G,\Omega)\quad \to \quad H^ 2(G,{\mathbb{C}}^*)\quad \to \quad 1 \] where the splitting is known to be natural in \(\Omega\). One result of this paper provides a natural splitting in G as well as \(\Omega\). If \(\Omega =K^*\) for a field K the author points out a connection with the problem of realizing complex projective representations of G and its subgroups over the field K. He considers also the case where the field \({\mathbb{C}}\) is replaced by any algebraically closed field. The results of this paper might also be useful in those parts of number theory where the group \(H^ 2(G,{\mathbb{C}}^*)\) is used.
Mathematics Subject Classification ID: 20C25
Mathematics Subject Classification ID: 20J05
zbMATH DE Number: 4130644
zbMATH Keywords: Schur multiplier
zbMATH Keywords: finite group
zbMATH Keywords: root of unity
zbMATH Keywords: universal coefficient theorem
zbMATH Keywords: split exact sequence
zbMATH Keywords: natural splitting
zbMATH Keywords: complex projective representations
reviewed by: Hans Opolka
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4130644