On groups with the same character degrees as almost simple groups with socle small Ree groups

Abstract

Let $G$ be a finite group and ${\rm cd}(G)$ denote the set of complex irreducible character degrees of $G$. In this paper, we prove that if $G$ is a finite group and $H$ is an almost simple group with socle $H_{0}= \, ^{2}{\rm G}_{2}(q)$, where $q=3^{f}$ with $f\geq 3$ odd such that ${\rm cd}(G)={\rm cd}(H)$, then $G$ is non-solvable and the chief factor $G'/M$ of $G$ is isomorphic to $H_{0}$. If, in particular, $f$ is coprime to $3$, then $G'$ is isomorphic to $H_{0}$ and $G/{\bf Z}(G)$ is isomorphic to $H$.