An answer to a question of Isaacs on character degree graphs

Let N be a normal subgroup of a finite group G. We consider the graph Gamma(GN)vertical bar whose vertices are the prime divisors of the degrees of the irreducible characters of G whose kernel does not contain N and two vertices are joined by an edge if the product of the two primes divides the degree of some of the characters of G whose kernel does not contain N. We prove that if Gamma(G vertical bar N) is disconnected then G/N is solvable. This proves a strong form of a conjecture of Isaacs. (c) 2004 Elsevier Inc. All rights reserved.