15.3 (2002)

Let $\alpha$ and $\beta$ be faithful non-linear irreducible characters of a finite group $G$. There are non-solvable groups $G$ giving examples when the product $\alpha\beta$ is again an irreducible character (for some of such $\alpha, \beta$). One example is $G = \text{SL}_2(5)$ with two irreducible characters of degree 2. In (I. Zisser, Israel J. Math., 84, no. 1–2 (1993), 147–151) it is proved that such an example exists in an alternating group $A_n$ if and only if $n$ is a square exceeding 4. But do solvable examples exist? Evidence (but no proof) that they do not is given in (I. M. Isaacs, J. Algebra, 223, no. 2 (2000), 630–646).