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

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.