Journal of Algebra and Its Applications Vol. 10, No. 2 (2011) 309–317 c  World Scientific Publishing Company DOI: 10.1142/S0219498811004598 QUASIRECOGNITION BY PRIME GRAPH OF THE SIMPLE GROUPS G 2 (q) AND 2 B 2 (q) * QINGLIANG ZHANG † , WUJIE SHI ‡ and RULIN SHEN § † School of Mathematical Sciences, Suzhou University Suzhou, Jiangsu, 215006, P. R. China ‡ School of Mathematics and Statics Chongqing University of Arts and Sciences Chongqing, 402160, P. R. China § Department of Mathematics, Hubei Institute for Nationalities Enshi, Hubei, 445000, P. R. China † qingliangstudent@163.com ‡ wjshi@suda.edu.cn § shenrulin@hotmail.com Received 28 December 2009 Accepted 11 May 2010 Communicated by L. Bokut 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(M), where M is G 2 (q)(q =3 2n+1 ) or 2 B 2 (q)(q =2 2n+1 > 2), then G is quasirecognizable by prime graph. Hence we generalize some known results of G 2 (q) and 2 B 2 (q). Keywords : Quasirecognition; finite simple group; prime graph. 1. Introduction Let n be an integer. 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 which is denoted by Γ(G) as follows: the 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 connected components of Γ(G) and let π 1 ,π 2 ,...,π t(G) be the connected components of Γ(G). If 2 ∈ π(G), we always suppose that 2 ∈ π 1 . And we denote by (a, b) the greatest common divisor of positive integers a and b. The spectrum of a finite group G, which is denoted by ω(G), is the set of its element orders. Obviously, ω(G) is partially ordered by divisibility. Therefore it is uniquely determined by μ(G), the subset of its maximal ∗ Project supported by the NNSF of China (Grant No. 10871032, No. 11026195) and the foundation of Educational Department of Hubei Province in China (No. Q20111901). ‡ Corresponding author. 309