The two-prime hypothesis: groups whose nonabelian composition factors are not isomorphic to ${{\mathrm{PSL}}}_2(q)$

Abstract

Let G be a finite group, and write ${\mathrm {cd}}(G)$ 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 $a, b \in {\mathrm {cd}}(G)$, 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 ${{\mathrm{PSL}}}_2(q)$ for any prime power q.