Huppert’s \(\rho - \sigma \) Conjecture

Here we look at Huppert’s \(\rho -\sigma \) conjecture for solvable groups, which posits the existence of certain “arithmetically large” character degrees of the solvable group G. We begin with a subtle variation of Gluck’s permutation lemma, and then use another large orbit theorem to prove the best currently known bound for Huppert’s conjecture for solvable groups.