On the torsion units in integral group rings of dihedral \(2\)-groups
type: Item
instance of: scholarly article
title: On the torsion units in integral group rings of dihedral \(2\)-groups
zbMATH Open document ID: 0830.16025
author: Alexander Zimmermann
published in: Journal of Algebra
publication date: 1995-11-28T00:00:00Z
review text: Let \(D_{2^n}\) be the dihedral group of order \(2^{n+1}\). The number of involutions in the group \(V(\mathbb{Z} D_{2^n})\) of units of augmentation 1 in the integral group ring \(\mathbb{Z} D_{2^n}\) is found to be \(1+2^n \cdot \prod^n_{k=1} h^+_{2^k}\), where \(h^+_{2^k}\) is the class number of the maximal real subfield of \(\mathbb{Q}(\zeta_{2^k})\), \(\zeta_{2^k}\) a primitive \(2^k\)-root of unity. The number of these involutions that belong to some group basis is \(1+2^n\). Hence, each involution belongs to a group basis if and only if each \(h^+_{2^k}=1\). Number theorists conjecture that this is indeed true. The interest in this is motivated by the result of Hughes and Pearson that \(\mathbb{Z} S_3\) has an involution of augmentation 1 that is not contained in any group basis. The proof, which uses class groups and Mayer-Vietoris sequences, also yields a description of the automorphism group of \(\mathbb{Z} D_{2^n}\). Finally, it is shown that the integral group ring of a semidihedral group of order at least 32 has an augmented involution that is not in any group basis.
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C10
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 783914
zbMATH Keywords: group bases
zbMATH Keywords: dihedral groups
zbMATH Keywords: number of involutions
zbMATH Keywords: units of augmentation 1
zbMATH Keywords: integral group rings
zbMATH Keywords: class numbers
zbMATH Keywords: class groups
zbMATH Keywords: Mayer- Vietoris sequences
zbMATH Keywords: automorphism groups
zbMATH Keywords: semidihedral groups
reviewed by: William H. Gustafson
MaRDI profile type: Publication
full work available at URL: jabr.1995.1178
OpenAlex ID: W2078733420
DOI: 10.1006/JABR.1995.1178
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description: scientific article; zbMATH DE number 783914