Projective special linear groups $PSL_4(q)$ are determined by the set of their character degrees

TL;DR

This paper confirms that the set of irreducible character degrees uniquely determines the structure of the projective special linear groups PSL_4(q) for q ≥ 13, supporting a broader conjecture about finite simple groups.

Contribution

It proves Huppert's conjecture for PSL_4(q), showing these groups are uniquely identified by their character degree sets among finite groups.

Findings

PSL_4(q) are characterized by their character degrees for q ≥ 13

Huppert's conjecture holds for this family of simple groups

Character degree sets determine the group structure in these cases

Abstract

Let $G$ be a finite group and let $cd(G)$ be the set of all irreducible complex character degrees of $G$. It was conjectured by Huppert in Illinois J. Math. 44 (2000) that, for every non-abelian finite simple group $H$, if $cd(G)=cd(H)$ then $G\cong H\times A$ for some abelian group $A$. In this paper, we confirm the conjecture for the family of projective special linear groups $\textrm{PSL}_4(q)$ with $q\geq 13$.