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.