n-RECOGNITION BY PRIME GRAPH OF THE SIMPLE GROUP PSL(2,q)

摘要

Let G be a finite group. The prime graph Gamma(G) of G is defined as follows. The vertices of Gamma(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge if there is an element in G of order pq. It is proved that if p > 11 and p not equivalent to 1 (mod 12), then PSL(2,p) is uniquely determined by its prime graph. Also it is proved that if p > 7 is a prime number and Gamma(G) = Gamma(PSL(2, p(2))), then G congruent to PSL(2, p2) or G congruent to PSL(2, p2). 2, the non-split extension of PSL(2, p(2)) by Z(2). In this paper as the main result we determine finite groups G such that Gamma(G) = Gamma(PSL(2, q)), where q = p(k). As a consequence of our results we prove that if q = p(k), k > 1 is odd and p is an odd prime number, then PSL(2,q) is uniquely determined by its prime graph and so these groups are characterizable by their prime graph.