Finite groups with three rational conjugacy classes (Q1692342)

\textit{G. Navarro} and \textit{P. H. Tiep} [Trans. Am. Math. Soc. 360, No. 5, 2443--2465 (2008; Zbl 1137.20009)] proved that for a finite group \(G\), \(|\mathrm{Irr}_{\mathbb{Q}}(G)|=1\) if and only if \(|\mathrm{Cl}_{\mathbb{Q}}(G)|=1\) and \(|\mathrm{Irr}_{\mathbb{Q}}(G)|=2\) if and only if \(|\mathrm{Cl}_{\mathbb{Q}}(G)|=2,\) where \(\mathrm{Irr}_{\mathbb{Q}}(G)\) denotes the set of irreducible \(\mathbb{Q}\)-characters of group \(G\) and \(\mathrm{Cl}_{Q}(G)\) denotes the set of \(\mathbb{Q}\)-classes of \(G.\) They conjunctured that for a finite group \(G,\) \(|\mathrm{Irr}_{\mathbb{Q}}(G)|=3\) if and only if \(|\mathrm{Cl}_{\mathbb{Q}}(G)|=3.\) In this paper, the author mentions that this is the best possible generalisation of the theorem of Navarro and Tiep [loc. cit.] and he proves one direction of this conjucture, i.e. for a finite group \(G\) holds \(|\mathrm{Irr}_{\mathbb{Q}}(G)|=3\) if \(|\mathrm{Cl}_{\mathbb{Q}}(G)|=3\). The proof uses the classification of finite simple groups.