Character codegrees, kernels, and Fitting heights of solvable groups

TL;DR

This work investigates how irreducible character codegrees relate to the Fitting height of solvable groups. It defines $cod(\chi)=|G:\ker(\chi)|/\chi(1)$ and proves a codegree analogue of the Broline–Garrison theorem: if $\ker(\chi)$ is not nilpotent, there exists $\xi$ with $\ker(\xi)<\ker(\chi)$ and $cod(\xi)>cod(\chi)$. As a consequence, for a nonidentity solvable group $G$, the Fitting height satisfies $\ell_{\mathbf{F}}(G)\le |\mathrm{cod}(G)|-1$, with sharper bounds $\ell_{\mathbf{F}}(G)\le \tfrac{1}{2}(|\mathrm{cod}(G)|+2)$ and $\ell_{\mathbf{F}}(G)\le 8\log_2(|\mathrm{cod}(G)|)+80$. The paper further studies the $p$-solvable case with a unique minimal normal subgroup, proving disjointness of $\mathrm{cod}(G/V)$ and $\mathrm{cod}(G|V)$ and leveraging classifications of solvable $1/2$-transitive groups to obtain the same two bounds on $\ell_{\mathbf{F}}(G)$. Overall, the results connect character codegrees to group structure, providing new upper bounds on the Fitting height and a dual perspective to classical kernel-based theorems.

Abstract

For an irreducible character $χ$ of a finite group $G$, let $\mathrm{cod}(χ):=|G: \ker(χ)|/χ(1)$ denote the codegree of $χ$, and let $\mathrm{cod}(G)$ be the set of irreducible character codegrees of $G$. In this note, we prove that if $\ker(χ)$ is not nilpotent, then there exists an irreducible character $ξ$ of $G$ such that $\ker(ξ)<\ker(χ)$ and $\mathrm{cod}(ξ)> \mathrm{cod}(χ)$. 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 $\ell_{\mathbf{F}}(G)\leq 8\log_2(|\mathrm{cod}(G)|)+80$.