Rationality and normal 2-complements.
Let \(G\) be a finite group, and denote by \(\text{Irr}_\mathbb{Q}(G)\) the set of rational valued irreducible characters of \(G\). The following results are proved: (i) Let \(P\) be a Sylow 2-subgroup of \(G\). Then all characters in \(\text{Irr}_\mathbb{Q}(G)\) are linear if and only if \(G\) has a normal 2-complement, every rational element of \(G\) is a 2-element and all characters in \(\text{Irr}_\mathbb{Q}(P)\) are linear. (ii) If all rational 2-elements of \(G\) are central in \(G\), then \(G\) has a normal 2-complement. The authors remark that the proof of the first result depends on CFSG, whereas the proof of the second only uses ordinary character theory.
scientific article; zbMATH DE number 5365174