A characterization of some finite simple groups by their character codegrees
Hung P. Tong-Viet
TL;DR
This work proves that for $H$ a finite simple exceptional group of Lie type or ${\rm PSL}_n(q)$ with $n\ge 4$, the set of character codegrees ${cod}(G)$ uniquely determines the group, i.e., ${cod}(G)={cod}(H)$ implies $G\cong H$. The authors employ a minimal counterexample framework where a unique minimal normal subgroup $N$ is shown to be elementary abelian, and they analyze whether $G$ is quasisimple or a nontrivial extension by $N$ using Clifford theory, Schur multipliers, and cohomology, together with powerful degree bounds (e.g., Landazuri–Seitz) and representation theory of classical groups. For ${\rm PSL}_n(q)$ with $n\ge 4$ they leverage Weil characters of ${\rm SL}_n(q)$ and cohomological vanishing to rule out non-split central extensions, ensuring $G\cong H$. In the symplectic case ${\rm PSp}_{2n}(q)$ they obtain a partial but substantial verification (including $n\ge 4$) and discuss remaining obstructions in the small-rank cases $n=2,3$ due to potential nontrivial $H^2$; overall the results extend codegree-determinacy to large families of Lie-type simple groups.
Abstract
For a finite group $G$ and an irreducible complex character $χ$ of $G$, the codegree of $χ$ is defined by $\textrm{cod}(χ)=|G:\textrm{ker}(χ)|/χ(1)$, where $\textrm{ker}(χ)$ is the kernel of $χ$. In this paper, we show that if $H$ is a finite simple exceptional group of Lie type or a projective special linear group and $G$ is any finite group such that the character codegree sets of $G$ and $H$ coincide, then $G$ and $H$ are isomorphic.