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Finite groups with three rational conjugacy classes. (English) Zbl 1390.20008

G. Navarro and 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.

MSC:

20C15 Ordinary representations and characters
20E45 Conjugacy classes for groups

Citations:

Zbl 1137.20009
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References:

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