The small Ree group $^{2}G_{2}(3^{2n+1})$ and related graph

Asboei, Alireza K. ; Amiri, Seyed S. S.

Commentationes Mathematicae Universitatis Carolinae, Tome 59 (2018) no. 3, pp. 271-276.

Résumé

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 $^{2}G_{2}(3^{2n+1})$.

Détail

DOI : 10.14712/1213-7243.2015.255

Classification : 05C25, 20D08
Keywords: main supergraph; simple Ree group; Thompson's problem