Irreducible Character Degrees and Normal Subgroups

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 ...

AFFINE SUBGROUPS OF THE CLASSICAL GROUPS AND THEIR CHARACTER DEGREES

In this paper we describe how the degrees of the irreducible characters of the affine subgroups of the classical groups under consideration can be found inductively. In [4] Gow obtained certain character degrees for all of the affine subgroups of the classical groups. We apply the method of Fischer to the above groups and, in addition to the character degrees given in [4], we obtain some ne...

Groups with Two Extreme Character Degrees and Their Normal Subgroups

We study the finite groups G for which the set cd(G) of irreducible complex character degrees consists of the two most extreme possible values, that is, 1 and |G : Z(G)|1/2. We are easily reduced to finite p-groups, for which we derive the following group theoretical characterization: they are the p-groups such that |G : Z(G)| is a square and whose only normal subgroups are those containing G′ ...

On Finite Groups With Two Irreducible Character Degrees

Prime Divisors of Irreducible Character Degrees

Let G be a finite group. We denote by ρ(G) the set of primes which divide some character degrees of G and by σ(G) the largest number of distinct primes which divide a single character degree of G. We show that |ρ(G)| ≤ 2σ(G) + 1 when G is an almost simple group. For arbitrary finite groups G, we show that |ρ(G)| ≤ 2σ(G) + 1 provided that σ(G) ≤ 2.