Character degree graphs and normal subgroups

Abstract

We consider the degrees of those irreducible characters of a group G G whose kernels do not contain a given normal subgroup N N . We show that if N ⊆ G ′ N \subseteq G’ and N N is not perfect, then the common-divisor graph on this set of integers has at most two connected components. Also, if N N is solvable, we obtain bounds on the diameters of the components of this graph and, in the disconnected case, we study the structure of N N and of G G .