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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 ∈ cd(G), the multiplicity of d in G, denoted by mG(d), is the number of irreducible characters of G having degree d. A finite group G is said to be a Tk-group for some integer k ≥ 1 if there exists a nontrivial degree d0 ∈ cd(G) such that mG(d0) = k and that for every d ∈ cd(G)− {1, d0}, the multiplicity of d in G is trivial, that is, mG(d) = 1. In this paper, we show that if G is a nonsolvable Tk-group for some integer k ≥ 1, then k = 2 and G ∼= PSL2(5) or PSL2(7).

Finite nonsolvable groups with many distinct character degrees