1 F eb 1 99 7 IRREDUCIBLE CHARACTER DEGREES AND NORMAL SUBGROUPS

1. Introduction. Let G be a finite group and, as usual, write cd(G) to denote the set of degrees of the irreducible characters of G. This set of positive integers encodes a great deal of information about the structure of G, and we mention just a few of the many known results of this type. If G is nilpotent (or more generally, if G is an M-group), a result of K. Taketa asserts that the derived length dl(G) is at most equal to |cd(G)|. (See Theorem 5.12 of [6].) It has been conjectured that the Taketa inequality dl(G) ≤ |cd(G)| holds for all solvable groups, but this remains unproved. (It is known, however, that that dl(G) ≤ 2|cd(G)| for all solvable groups G [4], and the Taketa inequality is known to hold if |G| is odd [1].) It is an old result of the first author that if |cd(G)| ≤ 3, then G is necessarily solvable and that dl(G) ≤ |cd(G)|. (See Theorems 12.15 and 12.6 of [6].) The Taketa inequality is also known to hold for solvable groups G if |cd(G)| = 4. (This result of S. Garrison is the principal theorem in his Ph.D. thesis [3], which remains unpublished.) If more information about the set cd(G) is known than its cardinality, then, of course, one can expect to be able to deduce correspondingly more information about G. For example, J. Thompson proved that if there is some prime p that divides every member of cd(G) exceeding 1, then G has a normal p-complement. (See Corollary 12.2 of [6].) Our goal in this paper is to study how the structure of a normal subgroup of G is influenced by the degrees of an appropriate subset of Irr(G). It seems reasonable that the characters that should be relevant to controlling the structure of N ⊳ G are exactly those whose kernels do not contain N , and so we introduce some convenient notation. Given that N ⊳ G, we write