# theorems of special linear group over a finite field

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 $\operatorname{SL}(2,\mathbb{F}_{3})$.

Title theorems of special linear group over a finite field TheoremsOfSpecialLinearGroupOverAFiniteField 2013-03-22 14:55:54 2013-03-22 14:55:54 Daume (40) Daume (40) 6 Daume (40) Theorem msc 20G15 ProjectiveSpecialLinearGroup