Characterization of some simple groups by the multiplicity pattern

Let be a finite group and let denote the set of all complex irreducible characters of Let be the set of all character degrees of For each positive integer the multiplicity of in is defined to be the number of irreducible characters of having the same degree The multiplicity pattern is the vector whose first coordinate is and for the th-coordinate of is the multiplicity of the th-smallest nontrivial character degree of In this paper, we show that every nonabelian simple group with at most distinct character degrees is uniquely determined by the multiplicity pattern.