Characters of $p’$-degree and Thompson’s character degree theorem

Abstract

Abstract. A classical theorem of John Thompson on character degrees asserts that if the degree of every ordinary irreducible character of a finite group G is 1 or divisible by a prime p, then G has a normal p-complement. We obtain a significant improvement of this result by considering the average of p ′ -degrees of irreducible characters. We also consider fields of character values and prove several improvements of earlier related results.

Methods and questions in character degrees of finite groups

“…For other related results, involving blocks, see Theorem C of [65] and Proposition 2.1 of [19]. The following result, essentially due to G. Navarro and P. H. Tiep [64], appears as Lemma 2.2 of [20] and takes into account fields of values. If N G and ϕ ∈ Irr(N) we write I G (ϕ) to denote the inertia subgroup of ϕ in G. Theorem 6.8 (Navarro-Tiep).…”

“…We will illustrate the relevance of both of these results in many problems. For instance, they have been used to obtain generalizations on the two most celebrated results on character degrees, the Itô-Michler Theorem and Thompson's Theorem (see [20,23,60] for instance). They were first used in conjunction in [49] in relation to Brauer's Problem 1 in his famous list of open problems in representation theory of finite groups [8].…”

Irreducible characters of even degree and normal Sylow 2-subgroups

“…Theorem 2.2 can be used to bound the number of irreducible characters of degrees 1 and 2 of finite groups with a nonabelian minimal normal subgroup, as shown in the next proposition. This should be compared with Proposition 3.2 of [9]. Proposition 2.3.…”