The two-prime hypothesis: groups whose nonabelian composition factors are not isomorphic to PSL2(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 is an element of 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 PSL2(q) for any prime power q.