Let G be a finite group. The main supergraph S(G) is a graph with vertex set G in which two vertices x and y are adjacent if and only if o(x) | o(y) or o(y) | o(x). In this paper, we will show that if S(G) ≅ S(S), where S belongs to a large class of finite non-solvable groups, then G ≅ S. This work is an important step in solving Thompson's problem.

We thank the referee for some extremely helpful remarks, which allowed us to improve the paper.