Nonsolvable groups satisfying the prime-power hypothesis

Let G   be a finite group with the degree set cd(G)cd(G) of its complex irreducible characters. We call that G   satisfies the prime-power hypothesis if, for distinct degrees χ(1),ψ(1)∈cd(G)χ(1),ψ(1)∈cd(G), the greatest common divisor gcd⁡(χ(1),ψ(1))gcd⁡(χ(1),ψ(1)) is a prime power. In this paper, we show that |cd(G)|⩽18|cd(G)|⩽18 if G is a nonsolvable group satisfying the prime-power hypothesis.