A CHARACTERIZATION OF PGL(2, p(n)) BY SOME IRREDUCIBLE COMPLEX CHARACTER DEGREES

For a finite group G, let cd(G) be the set of irreducible complex character degrees of G forgetting multiplicities and X-1(G) be the set of all irreducible complex character degrees of G counting multiplicities. Suppose that p is a prime number. We prove that if G is a finite group such that vertical bar G vertical bar = vertical bar PGL(2, p)vertical bar, p is an element of cd(G) and max(cd(G)) = p + 1, then G congruent to PGL(2, p), S L (2, p) or PSL(2, p) x Lambda, where Lambda is a cyclic group of order (2, p -1). Also, we show that if G is a finite group with X-1(G) = X-1(PGL(2, p(n))), then G congruent to PGL(2, p(n)). In particular, this implies that PGL(2, p n) is uniquely determined by the structure of its complex group algebra.