Bounding the number of irreducible character degrees of a group

M. Isaacs

Let \(G\) be a finite group and write \(cd(G)\) to denote the set of different degrees of the irreducible characters of \(G\). Also, let \(b(G)\) be the largest member of \(cd(G)\). Of course, \(|cd(G)| \leqslant b(G)\), but for some values of \(b(G),\) one can obtain very much stronger inequalities. For example, if \(b(G)\) is prime, then \(|cd(G)| \leqslant 4\). We discuss a generalization of this fact for solvable groups \(G\) for which \(b(G)\) has the form \(kp\), where \(p\) is prime and \(k < p\).