The solvability comes from a given set of character degrees (Q2816938)

Continuing earlier work of the authors et al. [Algebr. Represent. Theory 19, No. 2, 335--354 (2016; Zbl 1348.20007)] on the structure (particularly solvability) of finite groups with a special character degree set, the authors generalize and extend some these by proving the following theorem.NEWLINENEWLINE Theorem. Let \(G\) be a finite group and let \(a\), \(b\), and \(c\) be distinct integers greater than 1. Suppose that \(\operatorname{cd}(G)\subseteq\{1,a,b,c,ab,ac\}\). Then one of the following holds:{\parindent=7mm \begin{itemize}\item[(1)] \(G\) is solvable; \item[(2)] \(\operatorname{cd}(G)=\{1,a,b,c\}=\{1,9,10,16\}\); or \item[(3)] \(\operatorname{cd}(G)=\{1,a,b,c\}=\{1,q-1,q,q+1\}\) for some prime power \(q>3\).NEWLINENEWLINE\end{itemize}} The bulk of the proof consists of studying the character degrees of almost simple groups and depends on CFSG.