Quasirecognition by the prime graph of L3(q) where 3 < q < 100

Let G be a finite group. We construct the prime graph of G, which is denoted by Γ(G) as follows: the vertex set of this graph is the prime divisors of |G| and two distinct vertices p and q are joined by an edge if and only if G contains an element of order pq. In this paper, we determine finite groups G with Γ(G) = Γ(L3(q)), 2 ≤ q < 100 and prove that if q 6= 2, 3, then L3(q) is quasirecognizable by the prime graph, i.e., if G is a finite group with the same prime graph as the finite simple group L3(q), then G has a unique non-Abelian composition factor isomorphic to L3(q). As a consequence of our results we prove that the simple group L3(4) is recognizable and the simple groups L3(7) and L3(9) are 2−recognizable by the prime grap