Volume 32 • Number 3 • 2009   • Quasirecognition by Prime Graph of 2Dp(3) Where p = 2n + 1 ≥ 5 is a Prime A. Babai, B. Khosravi and N. Hasani Abstract. In this paper as the main result, we show that if is a finite group such that , where , is a prime number, then has a unique non-abelian composition factor isomorphic to . We also show that if is a finite group satisfying and , then . As a consequence of our result we give a new proof for a conjecture of W. J. Shi and J. X. Bi [A characteristic property for each finite projective special linear group, in Groups-Canberra 1989, 171-180, Lecture Notes in Math., 1456, Springer, Berlin] for . Application of this result to the problem of recognition of finite simple groups by the set of element orders are also considered. 2000 Mathematics Subject Classification: 20D05, 20D60. Full text: PDF