FINITE NONSOLVABLE GROUPS WITH MANY DISTINCT CHARACTER DEGREES

Let G be a finite group and let Irr(G) denote the set of all complex irreducible characters of G. Let cd(G) be the set of all character degrees of G. For a degree d is an element of cd(G), the multiplicity of d in G, denoted by m(G)(d), is the number of irreducible characters of G having degree d. A finite group G is said to be a T-k-group for some integer k >= 1 if there exists a nontrivial degree d(0) is an element of cd(G) such that m(G)(d(0)) = k and that for every d is an element of cd(G) - {1, d(0)}, the multiplicity of d in G is trivial, that is, m(G)(d) = 1. In this paper, we show that if G is a nonsolvable T-k-group for some integer k >= 1, then k = 2 and G congruent to PSL2(5) or PSL2(7).