Taketa’s theorem for some character degree sets

Abstract

Let cd(G) be the set of irreducible complex character degrees of a finite group G. The Taketa problem conjectures that if G is a finite solvable group, then $${{\rm dl}(G) \leqslant |{\rm cd} (G)|}$$ , where dl(G) is the derived length of G. In this note, we show that this inequality holds if either all nonlinear irreducible characters of G have even degrees or all irreducible character degrees are odd. Also, we prove that this inequality holds if all irreducible character degrees have exactly the same prime divisors. Finally, Isaacs and Knutson have conjectured that the Taketa problem might be true in a more general setting. In particular, they conjecture that the inequality $${{\rm dl}(N) \leqslant |{\rm cd} {(G \mid N)}|}$$ holds for all normal solvable subgroups N of a group G. We show that this conjecture holds if $${{\rm cd} {(G \mid N')}}$$ is a set of non-trivial p–powers for some fixed prime p.