Quasirecognition by prime graph of ²D_p(3) where p = 2ⁿ + 1 ≥ 5 is a prime

چکیده مقاله

In this paper as the main result, we show that if G is a finite group such that Γ(G) = Γ(²D_p(3)), where p = 2ⁿ +1, (n ≥ 2) is a prime number, then G has a unique non-abelian composition factor isomorphic to ²D_p(3). We also show that if G is a finite group satisfying {pipe}G{pipe} = {pipe}²D_p(3){pipe} and Γ(G) = Γ(²D_p(3)), then G = ²D_p(3). 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 ²D_p(3). Application of this result to the problem of recognition of finite simple groups by the set of element orders are also considered.