A criterion for a normal subgroup to be hypercentral based on class sizes

Abstract

Let $G$ be a finite group and $N$ a normal subgroup of $G$. We prove that the knowledge of the sizes of the conjugacy classes of $G$ that are contained in $N$ and of their multiplicities provides information of $N$ in relation to the structure of $G$. Among other results, we obtain a criterion to determine whether a Sylow $p$-subgroup of $N$ lies in the hypercentre of $G$ for a fixed prime $p$, and therefore, whether the whole subgroup $N$ is hypercentral in $G$.