SM-vanishing conjugacy classes of finite groups

Abstract

Let G be a finite group. In this paper, we say that g is an element of G is an SM-vanishing element of G, if there exists a strongly monolithic character chi of G such that chi(g) = 0. The conjugacy class of an SM-vanishing element of G is called an SM-vanishing conjugacy class of G. Our purpose here is to prove that for determining some properties of the structure of the group G, it is enough to consider the same arithmetical conditions on the sizes of SM-vanishing conjugacy classes of G instead of certain arithmetical conditions on the sizes of vanishing conjugacy classes of G.