The number of irreducible character degrees of solvable groups satisfying the one-prime hypothesis

Let G be a finite group, and write cd(G) for the set of degrees of irreducible characters of G. We say G satisfies the one-prime hypothesis if whenever a and b are distinct degrees in cd(G), then the greatest common divisor of a and b is either 1 or a prime. We show that if G is a solvable group satisfying the one-prime hypothesis, then vertical bar cd(G)vertical bar <= 9. We also construct a solvable group G satisfying the one-prime hypothesis with vertical bar cd(G)vertical bar = 9 which shows that the bound found in this paper is the best possible bound.