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main supergraph; simple Ree group; Thompson's problem
Let $G$ be a finite group. The main supergraph $\mathcal{S}(G)$ is a graph with vertex set $G$ in which two vertices $x$ and $y$ are adjacent if and only if $o(x) \mid o(y)$ or $o(y)\mid o(x)$. In this paper, we will show that $G\cong ^{2}G_{2}(3^{2n+1})$ if and only if $\mathcal{S}(G)\cong \mathcal{S}(^{2}G_{2}(3^{2n+1}))$. As a main consequence of our result we conclude that Thompson's problem is true for the small Ree group $^{2}G_{2}(3^{2n+1})$.
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