Irreducible Character Degree Sets of Solvable Groups

Abstract

In this paper we examine the following general question: Given a set X of positive integers that includes 1 is there a finite group G whose Ž Ž . Ž . < Ž .4. Ž character degree set cd G s x 1 x g Irr G equals X ? Technically, Ž . we should call cd G the set of irreducible character degrees, but these are the only character degrees that we will look at in this paper. Thus, when we say the character degrees of G, we mean the irreducible character . degrees . While this is a very interesting question, in order to answer it one must restrict the focus of the question. We restrict p to be a prime and m and n to be coprime integers that are greater than 1 and are not divisible 4 by p. We look at sets X that have the form X s 1 j P j M j N where 4 < < 4 P is a set of powers of p, M : m, mp with M G 1, and N : n, np with < < Ž . N G 1. We would like to know under what conditions X s cd G for some solvable group G. In this paper, we are able to give the following partial answer: