Character codegrees, kernels, and Fitting heights of solvable groups

Abstract For an irreducible character 𝜒 of a finite group 𝐺, let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>cod</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>χ</m:mi> <m:mo rspace="0.278em" stretchy="false">)</m:mo> </m:mrow> <m:mo rspace="0em">:</m:mo> <m:mo lspace="0em">=</m:mo> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>G</m:mi> <m:mo lspace="0.278em" rspace="0.278em">:</m:mo> <m:mi>ker</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>χ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mo>/</m:mo> <m:mi>χ</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \operatorname{cod}(\chi):=\lvert G:\ker(\chi)\rvert/\chi(1) denote the codegree of 𝜒, and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>cod</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \operatorname{cod}(G) be the set of irreducible character codegrees of 𝐺. In this note, we prove that if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ker</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>χ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \ker(\chi) is not nilpotent, then there exists an irreducible character 𝜉 of 𝐺 such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>ker</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ξ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>&lt;</m:mo> <m:mrow> <m:mi>ker</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>χ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> \ker(\xi)&lt;\ker(\chi) and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>cod</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ξ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>&gt;</m:mo> <m:mrow> <m:mi>cod</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>χ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> \operatorname{cod}(\xi)&gt;\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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="normal">ℓ</m:mi> <m:mi mathvariant="bold">F</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \ell_{\mathbf{F}}(G) does not exceed <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:mi>cod</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> \lvert\operatorname{cod}(G)\rvert-1 . Additionally, we provide two other upper bounds, <jats:di