Character codegrees, kernels, and Fitting heights of solvable groups

Abstract

For an irreducible character $\ddot{I}‡$ of a finite group $G$, let $\mathrm{cod}(\ddot{I}‡):=|G:\ker (\ddot{I}‡)|/\ddot{I}‡(1)$ denote the codegree of $\ddot{I}‡$, and let $\mathrm{cod}(G)$ be the set of irreducible character codegrees of $G$. In this note, we prove that if $\ker (\ddot{I}‡)$ is not nilpotent, then there exists an irreducible character $\hat{I}¾$ of $G$ such that $\ker (\hat{I}¾)<\ker (\ddot{I}‡)$ and $\mathrm{cod}(\hat{I}¾)>\mathrm{cod}(\ddot{I}‡)$. 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: ${\ell }_{\mathbf{F}}(G)\le \frac{1}{2}(|\mathrm{cod}(G)|+2)$, and ${\ell }_{\mathbf{F}}(G)\le 8{\log }_2(|\mathrm{cod}(G)|)+80$.