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

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 .