Nonsolvable normal subgroups and irreducible character degrees

Let the nonsolvable N be a normal subgroup of the finite group G and cd(G vertical bar N) denote the irreducible character degrees of G such that there exist respectively corresponding character kernels not containing N. Write vertical bar cd(G vertical bar N)vertical bar to stand for the cardinality of cd(G vertical bar N). Suppose that vertical bar cd(G vertical bar N)vertical bar <= 5. In this paper, we prove that if N is a minimal normal subgroup, then N is a simple group of Lie type. When N is a normal subgroup, we prove that G has a normal series 1 <= V < U <= N <= G such that V is solvable, U/V is a simple group of Lie type and the cardinality vertical bar cd(G/U)vertical bar <= 3. As an application, we investigate the structure of G when 5 <= vertical bar cd(G)vertical bar <= 6. Here cd(G) denotes the set of irreducible character degrees of G. (C) 2012 Elsevier Inc. All rights reserved.