A recent question of Gabriel Navarro asks whether it is true that the derived length a defect group less than or equal to number degrees irreducible characters in block. In this article, we bring new evidence towards validity statement.

In this article we prove the following result: that for any two natural numbers k and `, and for all sufficiently large symmetric groups Sn, there are k disjoint sets of ` irreducible characters of Sn, such that each set consists of characters with the same degree, and distinct sets have different degrees. In particular, this resolves a conjecture most recently made by Moretó in [5]. The method...