Classification of non-solvable groups with a given property. (Q2344078)

Inspired by the paper of \textit{A. Moreto, G. Qian} and \textit{W. Shi} [Arch. Math. 85, No. 2, 101-107 (2005; Zbl 1104.20027)], the authors treat the following classification problem for finite groups \(G\). Suppose \(G\) has the property \(|\Delta_p|\leq 4\) for any prime \(p\). Here \(\Delta_p\) stands for the subset of \(G\) with elements of order divisible by \(p\), and \(|\Delta_p|\) stands for the number of \(G\)-conjugacy classes contained in \(\Delta_p\). In this paper such non-solvable groups \(G\) have been determined (Theorem A). The proof of Theorem A depends on the classification of the finite simple groups.