QUASIRECOGNITION BY PRIME GRAPH OF Ln(2α) FOR SOME n AND α

Let G be a finite group. The prime graph of G is denoted by Gamma(G). In this paper as the main result, we show that if G is a finite group such that Gamma(G) = Gamma(L-n(2(alpha))), where n = 4m+3, 3 inverted iota alpha and a is odd, then G has a unique nonabelian composition factor isomorphic to L-n(2(alpha)). We also show that if G is a finite group satisfying vertical bar G vertical bar = vertical bar L-n(2(alpha))vertical bar and Gamma(G) = Gamma(L-n(2(alpha))), then G congruent to L-n(2(alpha)). As a consequence of our result we give a new proof for a conjecture of W. J. Shi and J. X. Bi for L-n(2(alpha)). Application of this result to the problem of recognition of finite simple groups by the set of element orders are also considered. Specially it is proved that by the above conditions, L-n(2(alpha)) is quasirecognizable by spectrum.