Let $\mathbb{F}_{q}$ be the finite field with $q$ elements, and consider the special linear group $\operatorname{SL}(n,\mathbb{F}_{q})$ over the field $\mathbb{F}_{q}$.

1. 1.

   $\operatorname{SL}(n,\mathbb{F}_{q})$ is finite. Furthermore, $\lvert\operatorname{SL}(n,\mathbb{F}_{q})\rvert=\frac{1}{q-1}\prod_{{i=0}}^{{n-1}}(q^{n}-q^{i})$.

2. 2.

   $\operatorname{SL}(n,\mathbb{F}_{q})$ is a perfect group, meaning that $[\operatorname{SL}(n,\mathbb{F}_{q}),\operatorname{SL}(n,\mathbb{F}_{q})]=\operatorname{SL}(n,\mathbb{F}_{q})$, where $[,]$ is the commutator bracket with two exceptions: $\operatorname{SL}(2,\mathbb{F}_{2})$ and $\operatorname{SL}(2,\mathbb{F}_{3})$.