\(SK_1\) of finite Abelian groups. I
type: Item
instance of: scholarly article
title: \(SK_1\) of finite Abelian groups. I
zbMATH Open document ID: 0576.18007
DOI: 10.1007/BF01394775
author: Roger C. Alperin
author: R. Keith Dennis
author: Michael R. Stein
published in: Inventiones Mathematicae
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143273
review text: For many years the Whitehead group of a group \(G\) had resisted computation. In the finite case, the portion that was so difficult to compute was \(SK_1(\mathbb{Z}[G])\). In this paper, it is shown that \[ SK_1(\mathbb{Z}[G])\simeq \prod_{p/| G|, p\neq 2} SK_1(\mathbb{Z}[G_p])^{q(G/G_p)} \times SK_1(\mathbb{Z}[G_2]) \times SK_1(\mathcal O[G_2])^{q(G/G_2)-1} \] where \(G_p\) is the Sylow \(p\)-subgroup of the finite group \(G\), \(q(H)\) is the number of irreducible irrational representations of a group \(H\), and \(\mathcal O\) is any totally imaginary ring of algebraic integers in which \(2\) is totally unramified. In the sequel [the authors et al., Invent. Math. 87, 253--302 (1987; Zbl 0605.18006)] \[ SK_1(\mathbb{Z}[G])\simeq \prod_{p/| G|}SK_1(\mathbb{Z}[G_p])^{q(G/G_p)} \times (\mathbb{Z}/2)^{(q(G/G_p)-1)(2^n-1-n-\binom{n}{2})}, \] where \(n\) is the order of \(G\). An efficient algorithm and many explicit computations are also given in the sequel.
reviewed by: Susan C. Geller
Mathematics Subject Classification ID: 19B99
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 3920756
zbMATH Keywords: special linear group
zbMATH Keywords: Whitehead group
zbMATH Keywords: \(SK_1(Z[G])\)
zbMATH Keywords: irreducible irrational representations
zbMATH Keywords: totally imaginary ring of algebraic integers
MaRDI profile type: Publication
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OpenAlex ID: W2883060349
description: scientific article; zbMATH DE number 3920756