The two-prime hypothesis: groups whose nonabelian composition factors are not isomorphic to PSL ₂(q)

Let G be a finite group, and write cd (G) for the degree set of the complex irreducible characters of G. The group G is said to satisfy the two-prime hypothesis if, for any distinct degrees a, b∈ cd (G) , the total number of (not necessarily different) primes of the greatest common divisor gcd (a, b) is at most 2. In this paper, we give an upper bound on the number of irreducible character degrees of nonsolvable groups satisfying the two-prime hypothesis and without composition factors isomorphic to PSL ₂(q) for any prime power q.