On the automorphism group of the integral group ring of \(S_ k \text{wr} S_ n\)
type: Item
instance of: scholarly article
title: On the automorphism group of the integral group ring of \(S_ k \text{wr} S_ n\)
zbMATH Open document ID: 0798.20003
DOI: 10.1016/0022-4049(92)90098-Z
published in: Journal of Pure and Applied Algebra
publication date: 1992-09-27T00:00:00Z
review text: \textit{S. K. Sehgal} [Topics in group rings (1978; Zbl 0411.16004)] conjectured that if \(G\) is a finite group and \(\theta\) a normalized automorphism of \(ZG\), the integral group ring of \(G\), then \(\theta\) is the composite of an automorphism of \(G\) and conjugation by a unit of \(QG\), the rational group algebra of \(G\). The author verifies this conjecture when \(G\) is the wreath product of the symmetric groups \(S_ k\) and \(S_ n\). If \(k=1\) this is a result of \textit{G. L. Peterson} [Proc. Am. Math. Soc. 59, 14-18 (1976; Zbl 0352.20004)], while if \(k=2\) the result was obtained by \textit{A. Giambruno, S. K. Sehgal}, and the author [Commun. Algebra 18, No. 6, 1989-2005 (1990; Zbl 0714.16020)]. In the proof the author shows that under \(\theta\) class sums map to class sums in a way that induces an automorphism of \(G\), and then applies a result of Sehgal to complete the theorem.
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20B25
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 20E22
zbMATH DE Number: 60686
zbMATH Keywords: finite group
zbMATH Keywords: normalized automorphism
zbMATH Keywords: integral group ring
zbMATH Keywords: wreath product
zbMATH Keywords: symmetric groups
author: Angela Valenti
reviewed by: E. Spiegel
MaRDI profile type: Publication
cites work: Automorphisms of the integral group ring of the hyperoctahedral group
cites work: Q3952291
cites work: Automorphisms of the Integral Group Ring of S n
cites work: Q4198893
full work available at URL: 0022-4049(92)90098-z
OpenAlex ID: W2054367255
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description: scientific article; zbMATH DE number 60686