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For a finite group G, let cd(G) be the set of irreducible complex character  degrees of G forgetting multiplicities and X1(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 |G| = |PGL(2,p) |, p  cd(G) and max(cd(G)) = p+1, then G  PGL(2,p), SL(2,  p) or PSL(2,p) x A, where A is a cyclic group of order (2, p-1). Also,  we show that if G is a finite group with X1(G) = X1(PGL(2,pn)), then G   PGL(2, pn). In particular, this implies that PGL(2, pn) is uniquely  determined by the structure of its complex group algebra.

A characterization of PGL (2,pn) by some irreducible complex character degrees