A note on codegrees and Taketa's inequality

TL;DR

This paper investigates the relationship between the derived length of finite groups and the set of codegrees of their irreducible characters, proposing a new inequality that holds in some cases and conjecturing its general validity for solvable groups.

Contribution

It establishes that the derived length is bounded above by the size of the codegree set in certain cases and conjectures this holds for all finite solvable groups.

Findings

Derived length is less than or equal to the size of the codegree set in some cases

Proposes a conjecture that this inequality holds for all finite solvable groups

Provides partial results supporting the conjecture

Abstract

Let $G$ be a finite group and ${\rm cd}(G)$ will be the set of the degrees of the complex irreducible characters of $G$. Also let ${\rm cod}(G)$ be the set of codegrees of the irreducible characters of $G$. The Taketa problem conjectures if $G$ is solvable, then ${\rm dl}(G) \leq |{\rm cd}(G)|$, where ${\rm dl}(G)$ is the derived length of $G$. In this note, we show that ${\rm dl}(G) \leq |{\rm cod}(G)|$ in some cases and we conjecture that this inequality holds if $G$ is a finite solvable group.