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

Abstract

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

10 pages