Nonsolvable groups with no prime dividing three character degrees

We consider nonsolvable finite groups G with the property that no prime divides at least three distinct character degrees of G. We first show that if S <= G <= AutS, where S is a nonabelian finite simple group, and no prime divides three degrees of G, then S congruent to PSL(2)(q) with q >= 4. Moreover, in this case, no prime divides three degrees of G if and only if G congruent to PSL(2)(q), G congruent to PGL(2)(q), or q is a power of 2 or 3 and G is a semi-direct product of PSL(2)(q) with a certain cyclic group. More generally, we give a characterization of nonsolvable groups with no prime dividing three degrees. Using this characterization, we conclude that any such group has at most 6 distinct character degrees, extending to the nonsolvable case the analogous earlier result of D. Benjamin for solvable groups. (C) 2011 Elsevier Inc. All rights reserved.