Two remarks on the Glauberman-Isaacs character correspondence.
Let \(G\) be a finite group and \(A\) acts on \(G\) as a group of automorphisms. Assume \((|A|,|G|)=1\). Then we have the Glauberman-Isaacs correspondence \(*\colon\text{Irr}(G)^A\to\text{Irr}(C_G(A))\) between the set of all irreducible characters of \(G\) invariant under \(A\) and the set of all irreducible characters of the centralizer of \(A\) in \(G\). \textit{G. Navarro} [J. Algebra 158, No. 2, 492-498 (1993; Zbl 0823.20007)] proved that if \(\alpha,\beta\in\text{Irr}(G)^A\) and \(\alpha\beta\in\text{Irr}(G)\), then \((\alpha\beta)^*=\alpha^*\beta^*\). One of the aims of the paper under review is to give a different proof of the above theorem. In another paper [Bull. Lond. Math. Soc. 26, No. 6, 513-522 (1994; Zbl 0829.20013)] \textit{G. Navarro} posed some open problems, one of them is: \(\chi^*(1)\mid\chi(1)\) for all \(\chi\in\text{Irr}(G)^A\). Another aim of the paper under review is to reduce the above conjecture to the case that \(G\) is a quasi-simple group.
Glauberman-Isaacs character correspondence
scientific article; zbMATH DE number 1997687