Abstract:
This work is a contribution to the classification of finite groups with an irreducible
character that vanishes on exactly one conjugacy class. Specifically, in this thesis we study finite non-solvable groups G that satisfy the above property when the character is primitive. We show that G has a homomorphic image that is either an almost simple group or a Frobenius group. We then classify all finite non-solvable groups with a faithful primitive irreducible character that vanishes on one conjugacy class. Our results answer two questions of Dixon and Rahnamai Barghi, one partially and the other completely.
A classical theorem of Burnside states that every irreducible character whose character degree is divisible by a prime number vanishes on at least one conjugacy class. Our results imply that if the degree of a primitive irreducible character of a finite group is divisible by two distinct primes, then the character vanishes on at least two conjugacy classes except when the group has a composition factor isomorphic to the Suzuki group Suz(8). This is an extension of Burnside's Theorem. Motivated by our result above, we show that for M-groups, groups of odd order and groups of derived length at most 3, if the character degree of an irreducible character of a group is divisible by two distinct
primes, then the irreducible character vanishes on at least two conjugacy classes.
For nilpotent groups, metabelian groups and groups whose distinct character degrees are pairwise relatively prime, we show that if the character degree of an irreducible character of a group is divisible by n distinct primes, then the irreducible character vanishes on at least n conjugacy classes for any positive integer n. This also holds when the group is solvable and the irreducible character is primitive.
