Siberian Mathematical Journal, Vol. 48, No. 3, pp. 570–577, 2007 Original Russian Text Copyright c  2007 Khosravi A. and Khosravi B. QUASIRECOGNITION BY PRIME GRAPH OF THE SIMPLE GROUP 2 G 2 (q) A. Khosravi and B. Khosravi UDC 519.542 Abstract: Let G be a finite group. The main result of this paper is as follows: If G is a finite group, such that Γ(G) = Γ( 2 G 2 (q)), where q =3 2n+1 for some n ≥ 1, then G has a (unique) nonabelian composition factor isomorphic to 2 G 2 (q). We infer that if G is a finite group satisfying |G| = | 2 G 2 (q)| and Γ(G) = Γ( 2 G 2 (q)) then G ∼ = 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. Keywords: quasirecognition, prime graph, simple group, element orders 1. Introduction If n is an integer then we denote by π(n) the set of all prime divisors of n. If G is a finite group then π(|G|) is denoted by π(G). We construct the prime graph of G as follows: The prime graph Γ(G) of a group G is the graph whose vertex set is π(G), and two distinct primes p and q are joined by an edge (we write p ∼ q) if and only if G contains an element of order pq. Let t(G) be the number of the connected components of Γ(G) and let π 1 (G),π 2 (G),...,π t(G) (G) be the connected components of Γ(G). Sometimes we use the notation π i instead of π i (G). If 2 ∈ π(G) then we always suppose that 2 ∈ π 1 . The connected components of nonabelian simple groups with at least two prime graph components are listed in [1, Tables 1a–1c]. The concept of prime graph arose in studying certain cohomological questions about the integral representations of finite groups. It was proved that for every finite group G we have t(G) ≤ 6 [2–4] and the diameter of Γ(G) is at most 5 (see [5]). Also, Hagie in [6] determined the finite groups G such that Γ(G) = Γ(S ) where S is a sporadic simple group. The set of element orders of G is denoted by π e (G). Obviously, π e (G) is partially ordered by divisibility. Therefore, it is uniquely determined by μ(G), the subset of its maximal elements. Denote by μ i (G) the set of those numbers n ∈ μ(G) whose all prime divisors belong to π i (G). Let G be a finite group such that G ∼ = H if and only if π e (G)= π e (H ). Then G is called recognizable by element orders (see [7–9]). A finite simple nonabelian group P is called quasirecognizable by element orders, if each finite group G with π e (G)= π e (P ) has a composition factor isomorphic to P (see [10]). Alongside we introduce similar concepts for the prime graph. Definition 1.1. A finite group G is called recognizable by prime graph if H ∼ = G for every finite group H with Γ(H ) = Γ(G). A finite simple nonabelian group P is called quasirecognizable by prime graph, if each finite group G with Γ(G) = Γ(P ) has a composition factor isomorphic to P . Alekseeva and Kondrat ′ ev in [10] proved that every finite simple group with at least three connected components (except A 6 ) is quasirecognizable by element orders. In the present article as the main result we prove the following theorem. Main Theorem. The simple group 2 G 2 (q) where q =3 2n+1 (n> 0) is quasirecognizable by prime graph. In this paper, all groups are finite and by simple groups we mean nonabelian simple groups. All further unexplained notations may be found in [11]. In the proof of the main theorem, we use the The second author was supported in part by a grant from IPM (No. 85200022). Tehran (Iran). Translated from Sibirski˘ ı Matematicheski˘ ı Zhurnal, Vol. 48, No. 3, pp. 707–716, May–June, 2007. Original article submitted October 27, 2005. Revision submitted February 9, 2006. 570 0037-4466/07/4803–0570 c  2007 Springer Science+Business Media, Inc.