Recognition of L2(q) by the Main Supergraph

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 G ∼= L2(q) if and only if S(G) ∼= S(L2(q)), where q is a prime power. This work implies that there is not a solvable group that has the same order type as the simple group L2(q).