Many authors were recently concerned with the following question: *What can be said about the structure of a finite group , if some information is known about the arithmetical structure of the degrees of the irreducible characters of ?*

Let be a finite group and be the set of all irreducible complex character degrees of counting multiplicities.

Let be an odd prime number and M = PGL*(2,p)*, M = *Z _{2}xPSL(2,p)* or
. In this paper we prove that is uniquely determined by its order and some information on its character degrees.
As a consequence of our results we prove that if is a finite group such that , then . This implies that is uniquely determined by the structure of its complex group algebra.

Key Words: Character degrees, order, projective special linear group, characterization

2010 Mathematics Subject Classification: Primary 20C15

Secondary 20D05, 20D60

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