THE CO-DEGREES OF IRREDUCIBLE CHARACTERS

Let G be a finite group. The co-degree of an irreducible character-chi of G is defined to be the number \G\/chi(1). The set of all prime divisors of all the co-degrees of the nonlinear irreducible characters of G is denoted by SIGMA(G). First we show that SIGMA(G) = pi(G) (the set of all prime divisors of \G\) unless G is nilpotent-by-abelian. Then we make SIGMA(G) a graph by adjoining two elements of SIGMA(G) if and only if their product divides a co-degree of some nonlinear character of G. We show that the graph SIGMA(G) is connected and has diameter at most 2. Additional information on the graph is given. These results are analogs to theorems obtained for the graph corresponding to the character degrees (by Manz, Staszewski, Willems and Wolf) and for the graph corresponding to the class sizes (by Bertram, Herzog and Mann). Finally, we investigate groups with some restriction on the co-degrees. Among other results we show that if G has a co-degree which is p-power for some prime p, then the corresponding character is monomial and O(p)(G) not-equal 1. Also we describe groups in which each co-degree of a nonlinear character is divisible by at most two primes. These results generalize results of Chillag and Herzog. Other results are proved as well.