Ree groups

Description

The Ree groups are two infinite families of finite groups of Lie type, found by Rimhak Ree.

1. For $q = 3^{2n+1}$ with $n \ge 0$, the group ${}^2G_2(q)$ is the subgroup of $G_2(\F_q)$ fixed by a twisted automorphism of characteristic $3$.
2. For $q = 2^{2n+1}$ with $n \ge 0$, the group ${}^2F_4(q)$ is the subgroup of $F_4(\F_q)$ fixed by a twisted automorphism of characteristic $2$.

Each twisted automorphism combines the exceptional graph automorphism of the Dynkin diagram with a field automorphism. Their orders are $$\abs{{}^2G_2(q)} = q^3 (q^3 + 1)(q - 1), \qquad \abs{{}^2F_4(q)} = q^{12} (q^6 + 1)(q^4 - 1)(q^3 + 1)(q - 1).$$ For $n \ge 1$ these groups are non-abelian simple groups.