On linear relations between character values
Finite groups \(G\) satisfying the following properties are characterized: There exist non-conjugate elements \(s\) and \(u\) of \(G\) and complex numbers \(a\), \(b\) and \(c\), \(ab \neq 0\), such that \(aX(s) +bX(u) = c\) is satisfied: (i) by every irreducible character of \(G\), or (ii) by every non-principal irreducible character of \(G\), but not by the principal character. In case (i), \(G\) turns out to be a solvable group of derived length at most 3. In case (ii) it is shown that if \(|G| > 3\), then either \(G\) is a Frobenius group of order \(2n\), \(n\) odd, with the kernel of order \(n\), or \(G \cong \text{PSL}(2,2^n)\), \(n \geq 2\).
scientific article; zbMATH DE number 3558044