On the number of irreducible real-valued characters of a finite group (Q2174540)

Let \(G\) be a finite group with at most \(k\) real-valued irreducible characters. In this paper, the authors show that there exists an integer-valued function of the variable \(k\) such that \(|G/\text{Sol}(G)|\leq f(k)\), where \(\text{Sol}(G)\) denotes the soluble radical of \(G\). In the case \(k=5\), they prove that \(G/\text{Sol}(G)\) is isomorphic to the trivial group, \(\text{SL}_3(2)\), \(\text{Alt}(5)\), \(\text{PSL}_2(8)\cdot 3\), or \(^2\text{B}_2(8)\cdot 3\).