SM-vanishing conjugacy classes of finite groups

Abstract

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