2-Frobenius \(\mathbb{Q}\)-groups. (Q1022919)

A finite group \(T\) is a 2-Frobenius group if it has a normal series \(1\triangleleft H\triangleleft K\triangleleft T\) where \(K/H\) is the kernel of the Frobenius group \(T/H\) and \(H\) the kernel of the Frobenius group \(K\); 2-Frobenius groups are always solvable. A finite group is called a \(\mathbb{Q}\)-group if any complex irreducible character of it is rational valued. In this paper 2-Frobenius \(\mathbb{Q}\)-groups are investigated. By \textit{R. Gow} [J. Algebra 40, 280-299 (1976; Zbl 0348.20017)] and \textit{P. Hegedűs} [Proc. Lond. Math. Soc., III. Ser. 90, No. 2, 439-471 (2005; Zbl 1071.20007)], the order of a 2-Frobenius \(\mathbb{Q}\)-group \(G\) is \(2^a3^b5^c\), with suitable \(a,b,c\geq 0\) and \(G\) can only contain a normal Sylow 5-subgroup. In this paper the following results are shown to be true for \(G\), one after the other: 1) \(K/H\cong C_3\) and any complement of \(K/H\) in \(G/H\) is isomorphic to \(C_2\); 2) If \(M\in\text{Syl}_3(G)\), then \(N_G(M)\cong S_3\) and \(C_G(M)=M\). 3) \(\zeta(G)=\{1\}\); 4) \([G,G]=K\) and \(G\) admits an irreducible character of degree 2; 5) the prime 5 does not divide the order of \(G\); 6) the order of \(G\) equals \(2^a3\) for some odd number \(a\); 7) \([[G,G],[G,G]]=[K,K]=H\); 8) there exists a 2-group \(N\trianglelefteq G\) with \(G/N\cong S_4\); 9) \(G\) contains a normal subgroup \(N\) with \(G/N\cong S_4\); 10) the subgroup \(N\) as in 9) is contained in \(H\); 11) in case \(H\) is a minimal normal subgroup of \(G\), then \(G\cong S_4\).