On groups with square-free gcd of character degree and codegree

Abstract: Let $G$ be a finite group and $\chi$ be an irreducible character of $G$. The codegree of $\chi$ is defined as $\chi^c(1) =\frac{|G: \ker\chi|}{\chi(1)}$. In a paper by Gao, Wang, and Chen \cite{GWC24}, it was shown that $G$ cannot satisfy the condition that $\gcd(\chi(1),\chi^c(1))$ is prime for all $\chi\in\Irr(G)^#$. We generalize this theorem by solving one of Guohua Qian's unsolved problems on character codegrees. In Qian's survey article \cite{Qiancodegree}, he inquires about the structure of non-solvable finite groups with square-free $\gcd$ instead. In particular, we prove that if $G$ is such that $\gcd(\chi(1),\chi^c(1))$ is square-free for every irreducible character $\chi$, then $G/\text{Sol}(G)$ is isomorphic to one among a particular list of almost simple groups.