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

Abstract

Let $G$ be a finite group and $\mathrm{cd}(G)$ 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 $H_0={}^2{\mathrm{G}}_2(q)$, where $q=3^f$ with $f\ge 3$ odd such that $\mathrm{cd}(G)=\mathrm{cd}(H)$, then $G$ is non-solvable and the chief factor $G^{\prime }/M$ of $G$ is isomorphic to $H_0$. If, in particular, $f$ is coprime to $3$, then $G^{\prime }$ is isomorphic to $H_0$ and $G/\mathbf{Z}(G)$ is isomorphic to $H$.

arXiv commentarXiv admin note: text overlap with arXiv:1601.06380