Quasirecognition by Prime Graph of ²Dp(3) Where p = 2ⁿ + 1 ≥ 5 is a Prime

In this paper as the main result, we show that if G is a finite group such that gamma(G) = gamma(2 Dp(3)), where p = 2n + 1, (n 2: 2) is a prime number, then G has a unique non-abelian composition factor isomorphic to 2 Dp(3). We also show that if G is a finite group satisfying IGI = 12 Dp(3)I and gamma(G) = gamma(2 Dp(3)), then G approximately equal to 2 Dp(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 finiteprojective special linear group, in Groups-Canberra 1989, 171-180, Lecture Notes in Math., 1456, Springer, Berlin] for 2 Dp(3) . Application of this result to the problem of recognition of finite simple groups by the set of element orders are also considered.