Character degrees of simple groups

Abstract

Gluck [2] and others have investigated the relationship between p(G), the set of primes dividing the degrees of the irreducible complex characters of G, and o(G), the greatest number of primes dividing the degree of a single irreducible character of G. when G is a finite solvable group. For such groups Gluck [2] has shown that 1 p(G)/ ,< (g(G))‘+ 100(G). In this paper we examine the relationship between these two quantities for G a finite nonabelian simple group. In order to state our results we need to introduce some notation. If G is a finite group and S is a subset of Irr(Gj, the set of irreducible characters of G, we will say S is a covering set of G if for every prime divisor p of / G 1 there is a character x in S such that p divides x( 1). If G has a covering set we will define the covering number of G, which we will denote by en(G), as the least number of elements in a covering set of G. Work of Michler [6, Theorem 3.31 has as a consequence that if G is a nonabelian simple group then G has a covering set; in other words, p(G) = n(G), where n(G) is the set of prime divisors of the order of G. For such groups then en(G) ,( 1 r(G)1 However, more is true.