Automorphisms of the integral group ring of the wreath product of a \(p\)- group with \(S_ n\)
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 \(G\).
scientific article; zbMATH DE number 219336