A character theoretic criterion for Fitting height

Abstract For a finite group 𝐺, set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable columnspacing="0pt" displaystyle="true" rowspacing="0pt"> <m:mtr> <m:mtd columnalign="right"> <m:mrow> <m:msub> <m:mi>Irr</m:mi> <m:mi>N</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:mtd> <m:mtd columnalign="left"> <m:mrow> <m:mo rspace="0.278em">:=</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mi>χ</m:mi> <m:mo>∈</m:mo> <m:mi>Irr</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo lspace="0em" rspace="0.167em">∣</m:mo> <m:mi>χ</m:mi> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mo>∤</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: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:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="right"> <m:mrow> <m:msub> <m:mi>cd</m:mi> <m:mi>N</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:mtd> <m:mtd columnalign="left"> <m:mrow> <m:mrow> <m:mi/> <m:mo lspace="0.278em" rspace="0.278em">:=</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:mi>χ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo fence="true" lspace="0em" rspace="0em">∣</m:mo> <m:mrow> <m:mi>χ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msub> <m:mi>Irr</m:mi> <m:mi>N</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:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> <m:mo lspace="0em">.</m:mo> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:math> \begin{aligned} \operatorname{Irr}_{N}(G)&amp;:=\{\chi\in\operatorname{Irr}(G)\mid\chi{(1)^{2}}\nmid\lvert G:\operatorname{ker}\chi\rvert\},\\ \operatorname{cd}_{N}(G)&amp;:=\{\chi(1)\mid\chi\in\operatorname{Irr}_{N}(G)\}.\end{aligned} In this paper, we study finite groups 𝐺 with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:msub> <m:mi>cd</m:mi> <m:mi>N</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:mo stretchy="false">|</m:mo> </m:mrow> <m:mo>≤</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> \lvert\operatorname{cd}_{N}(G)\rvert\leq 1 . In particular, we prove that if 𝐺 has odd order, then <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>h</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>≤</m:mo> <m:mrow> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:msub>