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. $\operatorname{SL}(n, \mathbb F_q)$ 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. $\operatorname{SL}(n, \mathbb F_q)$ 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 $\operatorname{SL}(2,\mathbb{F}_3)$ .