Non-abelian composition factors of \(m\)-rational groups (Q2357525)

Let \(m\) be a positive integer. A finite group \(G\) is \(m\)-rational if \(| \mathbb{Q}(\chi): \mathbb{Q} | \, \big | \, m\) for every irreducible complex character \(\chi\) of \(G\). \textit{W. Feit} and \textit{G. M. Seitz} [Ill. J. Math. 33, No. 1, 103--131 (1989; Zbl 0701.20005)] classified the finite simple rational (that is 1-rationals) groups. In this paper, the author proves the following result (Theorem 1.3): Let \(G\) be a finite \(m\)-rational group. Then, any non-abelian composition factor \(S\) of \(G\) is either an alternating group \(A_{n}\), a sporadic group, or \(S\) belongs to a finite set of groups of Lie type \(\mathscr{F}(m)\). In the case \(G \in \mathscr{F}(m)\), we may bound the order of \(S\) in terms of \(m\). Furthermore, the author provides a complete classification of the simple groups that may occur in a composition factor of quadratic groups (i.e. \(m=2\)). They can be alternating groups \(A_{n}\) (\(n \geq 5\)), one of the 20 sporadic groups from the following list: \(M_{11}\), \(M_{12}\), \(M_{22}\), \(M_{23}\), \(M_{24}\), \(J_{2}\), \(\mathrm{Co}_{1}\), \(\mathrm{Co}_{2}\), \(\mathrm{Co}_{3}\), \(\mathrm{Fi}_{22}\), \(\mathrm{Fi}_{23}\), \(\mathrm{Fi}_{24}'\), HS, McL, He, Suz, HN, Th, \(B\), \(M\) or one of the following groups of Lie type: \(L_{2}(q)\) (\(q \in \{7,8,11,16,27\}\)), \(L_{3}(3)\), \(L_{3}(4)\), \(L_{4}(3)\), \(U_{3}(q)\) (\(q \in \{3,4,5,8\}\)), \(U_{4}(2)\), \(U_{4}(3)\), \(U_{5}(2)\), \(U_{5}(4)\), \(U_{6}(2)\), \(S_{4}(4)\), \(S_{6}(2)\), \(S_{6}(3)\), \(S_{8}(2)\), \(O_{7}(2)\), \(O_{8}^{+}(2)\), \(O_{8}^{+}(3)\), \(O_{8}^{-}(2)\), \(O_{10}^{-}(2)\), \(^{2}\!E_{6}(2)\), \(F_{4}(2)\), \(^{2}\!F_{4}(2)'\), \(G_{2}(3)\), \(G_{2}(4)\), \(^{3}\!D_{4}(2)\).