The Simple Ree groups ${}^2F_4(q^2)$ are determined by the set of their character degrees

TL;DR

This paper proves that the set of character degrees uniquely determines the simple Ree group ${}^2F_4(q^2)$ up to direct product with an abelian group, confirming Huppert's Conjecture for these groups.

Contribution

It establishes that simple Ree groups ${}^2F_4(q^2)$ are uniquely identified by their character degree sets, verifying a longstanding conjecture for these groups.

Findings

${}^2F_4(q^2)$ groups are characterized by their character degrees

Huppert's Conjecture holds for ${}^2F_4(q^2)$ when $q^2 extgreater=8$

Any group with the same character degrees as ${}^2F_4(q^2)$ is a direct product with an abelian group

Abstract

Let $G$ be a finite group. Let ${\rm{cd}}(G)$ be the set of all complex irreducible character degrees of $G.$ In this paper, we will show that if ${\rm{cd}}(G)={\rm{cd}}(H),$ where $H$ is the simple Ree group ${}^2F_4(q^2),q^2\geq 8,$ then $G\cong H\times A,$ where $A$ is an abelian group. This verifies Huppert's Conjecture for the simple Ree groups ${}^2F_4(q^2)$ when $q^2\geq 8.$