Abstract: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 $^{2}G_{2}(3^{2n+1})$.
Keywords: main supergraph; simple Ree group; Thompson's problem
DOI: DOI 10.14712/1213-7243.2015.255
AMS Subject Classification: 20D08 05C25