Sylow 2-subgroups of rational solvable groups.
By definition a finite group \(G\) is called rational if every element \(g\) in \(G\) is conjugate to all elements \(g^m\) where \(m\) is coprime to \(|G|\). This is equivalent to the fact that every irreducible character of \(G\) is rational valued. An old conjecture stated by \textit{D. Kletzing}, [Structure and representations of \(\mathbf Q\)-groups. Lect. Notes Math. 1084. Berlin: Springer-Verlag (1984; Zbl 0546.20005)] proposes that if \(G\) is a rational group, then a Sylow 2-subgroup of \(G\) should be a rational group, too. This conjecture appears again in The Kourovka notebook [\textit{V. D. Mazurov} (ed.) and \textit{E. I. Khukhro} (ed.), The Kourovka note book. Unsolved problems in group theory. Novosibirsk: Institute of Mathematics, Russian Academy of Sciences, Siberian Div. (2006; Zbl 1084.20001)] as question number 16.36. In the paper under review the authors prove that the conjecture is false by finding two counterexamples of order 1536. As a matter of fact they located these two groups using both group theory computer packages GAP and Magma and showed that they are rational groups but their Sylow 2-subgroups have 53 irreducible characters with only 2 non-rational irreducible characters.
scientific article; zbMATH DE number 6116845