Brauer characters and fields of values
Let \(G\) be a finite group, \(p\) a prime number, and \(\mathbb F\) a subfield of the complex numbers. A \(p\)-regular element \(x\) of a finite group \(G\) is called an \(\mathbb F\)-element iff \(\phi (x)\in \mathbb F\) for every \(p\)-Brauer character \(\phi\) of \(G\). Two results are proved, which are modular analogues of results of \textit{I. M. Isaacs} and \textit{G. Navarro} [J. Group Theory 12, No. 5, 635--650 (2009; Zbl 1186.20009)], as follows. First, if \(N\) is a normal subgroup of \(G\) such that the identity is the only \(p\)-regular \(\mathbb F\)-element of \(N\), then the canonical epimorphism \(G \rightarrow G/N\) induces a bijection between the conjugacy classes of \(p\)-regular \(\mathbb F\)-elements of \(G\) and the conjugacy classes of \(p\)-regular \(\mathbb F\)-elements of \(G/N\). Second, every finite group \(G\) has a unique largest normal subgroup \(N\) such that the identity is the only \(p\)-regular \(\mathbb F\)-element of \(N\). The proof of the injectivity in the first result constitutes the bulk of the work in the paper.
scientific article; zbMATH DE number 6803034
scientific article