On vanishing criteria that control finite group structure

Abstract

Many results have been established that show how arithmetic conditions on conjugacy class sizes affect group structure. A conjugacy class in $G$ is called vanishing if there exists some irreducible character of $G$ which evaluates to zero on the conjugacy class. The aim of this paper is to show that for some classical results it is enough to consider the same arithmetic conditions on the vanishing conjugacy classes of the group.