Character codegrees, kernels, and Fitting heights of solvable groups

TL;DR

This paper explores the relationship between character codegrees and the structure of solvable groups, establishing bounds on their Fitting height based on the set of codegrees and extending classical theorems.

Contribution

It introduces a codegree analogue of a classical theorem, linking character kernels and codegrees, and provides new bounds on the Fitting height of solvable groups.

Findings

Existence of characters with larger codegrees and smaller kernels when kernels are not nilpotent.

Fitting height of a nonidentity solvable group is at most one less than the size of its codegree set.

New upper bounds for Fitting height based on codegree set size and logarithmic functions.

Abstract

For an irreducible character $\chi$ of a finite group $G$, let $\mathrm{cod}(\chi):=|G: \ker(\chi)|/\chi(1)$ denote the codegree of $\chi$, and let $\mathrm{cod}(G)$ be the set of irreducible character codegrees of $G$. In this note, we prove that if $\ker(\chi)$ is not nilpotent, then there exists an irreducible character $\xi$ of $G$ such that $\ker(\xi)<\ker(\chi)$ and $\mathrm{cod}(\xi)> \mathrm{cod}(\chi)$. This provides a character codegree analogue of a classical theorem of Broline and Garrison. As a consequence, we obtain that for a nonidentity solvable group $G$, its Fitting height $\ell_{\mathbf{F}}(G)$ does not exceed $|\mathrm{cod}(G)|-1$. Additionally, we provide two other upper bounds for the Fitting height of a solvable group $G$ as follows: $\ell_{\mathbf{F}}(G)\leq \frac{1}{2}(|\mathrm{cod}(G)|+2)$, and $\ell_{\mathbf{F}}(G)\leq 8\log_2(|\mathrm{cod}(G)|)+80$.