Groups whose non-linear irreducible characters are rational valued.
Let \(G\) be a finite group. \(G\) is called a rational group or \(\mathbb{Q}\)-group if all irreducible characters of \(G\) are rational valued, and \(G\) is called a \(\mathbb{Q}_1\)-group if all nonlinear irreducible characters of \(G\) are rational valued. The paper under review presents examples of nonabelian \(\mathbb{Q}_1\)-groups which are not \(\mathbb{Q}\)-groups and also presents some results on the structure of \(\mathbb{Q}_1\)-groups. For example, if \(G\) is nonabelian, then \(G\) is a \(\mathbb{Q}_1\)-group if and only if \(\chi(g)\in\mathbb{Q}\) for all \(\chi\in\text{Irr}(G)\) and all \(g\in V(G)\). (Here \(V(G)\) denotes the vanishing-off subgroup of \(G\) which is generated by all \(g\in G\) such that there is a nonlinear \(\chi\in\text{Irr}(G)\) with \(\chi(g)\neq 0\).) Combining this with a result of M. Lewis that \(G=V(G)\) for nonsolvable groups shows that the notions of \(\mathbb{Q}\)-group and \(\mathbb{Q}_1\)-group are the same for nonsolvable groups.
scientific article; zbMATH DE number 5716891