Group of Lie type

Remarks

The group $\mathbf{G}^F$ is always finite. Example: take $\mathbf{G} = \mathrm{GL}_n(k)$ and let $F$ raise every matrix entry to the power $q$, where $q$ is a power of $p$. Then $F$ is the Frobenius endomorphism for the standard structure over $\F_q$, and $\mathbf{G}^F = \mathrm{GL}_n(\F_q)$, since the elements of $k$ fixed by $x \mapsto x^q$ are exactly the elements of $\F_q$. Non-example: $\mathrm{GL}_n(\R)$ for $n \ge 1$ is not a finite group of Lie type, since it is infinite. Many authors also call the simple groups obtained from $\mathbf{G}^F$, such as $\mathrm{PSL}_n(\F_q)$, groups of Lie type. The Chevalley groups over $\F_q$ come from the case where $\mathbf{G}$ is split over $\F_q$ and $F$ is the corresponding Frobenius endomorphism. The twisted groups, such as the unitary groups, come from the other Steinberg endomorphisms. For the Suzuki groups and the Ree groups, the map $F$ is not a Frobenius endomorphism, but $F^2$ is one.