Is it true that for any finite nontrivial group G, there exist two inequivalent irreducible representations of G over the complex numbers that have the same degree.
If so, is there an easy proof? If not, what is the smallest counterexample?
Note: Any counterexample group must be perfect, because if the abelianization is nontrivial, we get multiple irreducible representations of degree one. [EDIT: Further, as Colin Reid notes in the comment, a minimal counterexample must be a simple (non-abelian) group]. This whittles down our search considerably. The general expressions for the degrees of irreducible representations for the families of simple groups that I've checked suggests that there is plenty of repetition of degrees in these cases.