On a question of Dixon and Rahnamai Barghi

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 $\rm{A}_{n}$, $n\geq 5$ or $\rm{PSL}_{2}(q)$, where $q\geq 4$ is a prime power, when $\chi$ is faithful. Our results partially answer a question of Dixon and Rahnamai Barghi.