On a question of Dixon and Rahnamai Barghi

Abstract

Let $G$ be a finite non-solvable group with a primitive irreducible character $\chi$ that vanishes on one conjugacy class. We show that $G$ has a homomorphic image that is either almost simple or a Frobenius group. We also classify such groups $G$ with a composition factor isomorphic to a sporadic group, an alternating group ${\mathrm{A}}_{\mathrm{n}}$, $n\ge 5$ or ${\mathrm{PSL}}_2(\mathrm{q})$, where $q\ge 4$ is a prime power, when $\chi$ is faithful. Our results partially answer a question of Dixon and Rahnamai Barghi.

13 pages