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 $ \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.