On the automorphism group of the integral group ring of \(S_ k \text{wr} S_ n\)
\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.
scientific article; zbMATH DE number 60686
scientific article