Character codegrees, kernels, and Fitting heights of solvable groups

Abstract For an irreducible character 𝜒 of a finite group 𝐺, let cod ( χ ) : = | G : ker ( χ ) | / χ ( 1 ) \operatorname{cod}(\chi):=\lvert G:\ker(\chi)\rvert/\chi(1) denote the codegree of 𝜒, and let cod ⁡ ( G ) \operatorname{cod}(G) be the set of irreducible character codegrees of 𝐺. In this note, we prove that if ker ⁡ ( χ ) \ker(\chi) is not nilpotent, then there exists an irreducible character 𝜉 of 𝐺 such that ker ⁡ ( ξ ) < ker ⁡ ( χ ) \ker(\xi)<\ker(\chi) and cod ⁡ ( ξ ) > cod ⁡ ( χ ) \operatorname{cod}(\xi)>\operatorname{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 ℓ F ⁢ ( G ) \ell_{\mathbf{F}}(G) does not exceed | cod ⁡ ( G ) | − 1 \lvert\operatorname{cod}(G)\rvert-1 . Additionally, we provide two other upper bounds,