Authors’ summary: “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 \(S\leqslant G\leqslant\operatorname{Aut}\,S\), where \(S\) is a nonabelian finite simple group, and no prime divides three degrees of \(G\), then \(S\cong\text{PSL}_2(q)\) with \(q\geqslant 4\). Moreover, in this case, no prime divides three degrees of \(G\) if and only if \(G\cong\text{PSL}_2(q)\), \(G\cong\text{PGL}_2(q)\), or \(q\) is a power of 2 or 3 and \(G\) is a semi-direct product of \(\text{PSL}_2(q)\) 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.”
Indeed, that summary covers most of the paper. In order to establish what the authors are aiming at, they work with a lot of details with regard to character degrees and its graphs. The reader should take notice of all the rich details in this paper. Implicit the authors need much specific facts from number theory which they work out very nicely.