Quasirecognition by prime graph of the simple group 2G2(q)

Let G be a finite group. The main result of this paper is as follows: If G is a finite group. such that Gamma(G) = Gamma((2)G(2)(q)), where q = 3(2n+1) for some n >= 1, then G has a (unique) nonabelian composition factor isoinorphic to (2)G(2)(q). We infer that if G is a finite group satisfying vertical bar G vertical bar = vertical bar(2)G(2)(q)vertical bar and Gamma(G) = Gamma((2)G(2)(q)) then G similar or equal to (2)G(2)(q). This enables us to give new proofs for some theorems; e.g., a conjecture of W. Shi and J. Bi. Some applications of this result are also considered to the problem of recognition by element orders of finite groups.