Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(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 nequiv 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 Γ(G) = Γ(PSL(2,p^{2})), then G ≅ PSL(2,p^{2}) or G ≅ PSL(2,p^{2}).2, the nonsplit extension of PSL(2,p^{2}) by \mathbbZ_{2}� In this paper as the main result we determine finite groups G such that Γ(G) = Γ(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.
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