Nonsolvable normal subgroups and irreducible character degrees

Abstract

Let the nonsolvable N be a normal subgroup of the finite group G and cd(G|N) denote the irreducible character degrees of G such that there exist respectively corresponding character kernels not containing N. Write |cd(G|N)| to stand for the cardinality of cd(G|N). Suppose that |cd(G|N)|⩽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 |cd(G/U)|⩽3. As an application, we investigate the structure of G when 5⩽|cd(G)|⩽6. Here cd(G) denotes the set of irreducible character degrees of G.