Ree groups and Moufang octagons

Question

Consider a Ree group of type $^2\mathrm{F}_4$, defined over the field $k$. Tits showed that every Moufang generalized octagon arises as a natural geometric module on which a Ree group of this type acts.

Suppose that $G$ and $H$ are Ree groups of type $^2\mathrm{F}_4$, respectively over the fields $k$ and $\ell$, such that $k$ is a subfield of $\ell$.

• Does this imply that $G$ is a subgroup of $H$ ? (It seems natural to think that it does.) Is there an easy way to see this ?

• And if the answer is "yes," is the generalized octagon corresponding to $G$ a suboctagon of the octagon corresponding to $H$ ?

Answer 1

Only having $k$ as subfield of $\ell$ is not sufficient (for both questions). The Ree groups (and the generalized octagons) are determined by a pair $(k, \theta)$, where $k$ is a field of characteristic $2$, and $\theta$ is a so-called Tits endomorphism of $k$, i.e., an endomorphism with $\theta^2 = \operatorname{Frob}$.

A given field $k$ with $\operatorname{char}(k)=2$ might or might not admit a Tits endomorphism, and (importantly for your questions) if it does, it might or might not be unique. It thus makes sense to define $$ (k, \theta) \leq (\ell, \rho) \iff k \leq \ell \text{ and } \rho_{|k} = \theta. $$ This is the condition that you need in order to ensure an affirmative answer to both of your questions.

If you want an explicit example, you can think of fields of the form $\mathbb{F}_2(\alpha, \beta, \gamma, \delta)$, and you can construct Tits endomorphisms $\theta_1$ mapping $\alpha$ to $\beta$, $\beta$ to $\alpha^p$, $\gamma$ to $\delta$, $\delta$ to $\gamma^p$ and $\theta_2$ mapping $\alpha$ to $\gamma$, $\gamma$ to $\alpha^p$, $\beta$ to $\delta$, $\delta$ to $\beta^p$.