Characterization of some projective special linear groups by character degreee graph and order

The character degree graph of a finite group G is the graph whose vertices are the prime divisors of the irreducible character degrees of G and two vertices p and q are joined by an edge if pq divides some irreducible character degree of G. In this talk we prove that the simple groups PSL(2; p3) and PSL(2; p4), where p 11 is a prime number, are uniquely determined by their character degree graphs and their orders. Let 1(G) be the set of all irreducible complex character degrees of G counting multiplicities. As a consequence of our results we prove that if G is a finite group such that 1(G) = 1(PSL(2; q)), where q = p3 or q = p4 and p 11 is a prime, then G PSL(2; q). This implies that PSL(2; q) is uniquely determined by the structure of its complex group algebra.