Automorphisms of the integral group ring of the wreath product of a \(p\)- group with \(S_ n\)
type: Item
instance of: scholarly article
title: Automorphisms of the integral group ring of the wreath product of a \(p\)- group with \(S_ n\)
zbMATH Open document ID: 0796.16030
author: Antonio Giambruno
author: Sudarshan K. Sehgal
published in: Rocky Mountain Journal of Mathematics
publication date: 1993-06-29T00:00:00Z
review text: If \(\theta\) is an automorphism of \(\mathbb{Z} G\), the integral group ring of the finite group \(G\), a conjecture, which the authors attribute to Zassenhaus, states that \(\theta\) is the composite of an automorphism of \(G\) and conjugation by a unit of \(\mathbb{Q} G\), the rational group algebra. When \(G\) is the wreath product of a finite group \(H\) and \(S_ n\), the symmetric group on \(n\) symbols, this conjecture has been verified if \(H\) is abelian by the authors and \textit{A. Valenti} [Commun. Algebra 19, 519- 534 (1991; Zbl 0727.16015)], and by \textit{A. Valenti}, when \(H\) is a symmetric group [J. Pure Appl. Algebra 78, No. 2, 203-211 (1992)]. In this paper the authors prove the conjecture when \(H\) is a \(p\)-group by showing that \(\theta\) maps class sums to class sums in such a way as to induce an automorphism \(\sigma\) of \(G\). The mapping \(\sigma\) is first defined on \(H^ n\) using a result of Weiss on matrices over the \(p\)-adic group ring and then extended to all of \(G\).
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 20E22
zbMATH DE Number: 219336
zbMATH Keywords: unit of the rational group algebra
zbMATH Keywords: automorphism
zbMATH Keywords: integral group ring
zbMATH Keywords: finite group
zbMATH Keywords: wreath product
zbMATH Keywords: symmetric group
zbMATH Keywords: \(p\)-group
zbMATH Keywords: class sums
zbMATH Keywords: matrices over the \(p\)-adic group ring
reviewed by: E. Spiegel
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072657
OpenAlex ID: W2090206217
cites work: Automorphisms of the integral group rings of some wreath products
cites work: Q3952291
cites work: Automorphisms of the Integral Group Ring of S n
cites work: Q4198893
cites work: On the automorphism group of the integral group ring of \(S_ k \text{wr} S_ n\)
cites work: Rigidity of \(p\)-adic \(p\)-torsion
DOI: 10.1216/RMJM/1181072657
description: scientific article; zbMATH DE number 219336