Let $G$ be a finite group. In this paper, we say that $g \in G$ is an SM-vanishing element of $G$ , if there exists a strongly monolithic character $χ$ of $G$ such that $χ (g) = 0$ . 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 $G$ .

Communicated by M. L. Lewis

Keywords:

AMSC: 20C15

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