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.

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

Title	theorems of special linear group over a finite field
Canonical name	TheoremsOfSpecialLinearGroupOverAFiniteField
Date of creation	2013-03-22 14:55:54
Last modified on	2013-03-22 14:55:54
Owner	Daume (40)
Last modified by	Daume (40)
Numerical id	6
Author	Daume (40)
Entry type	Theorem
Classification	msc 20G15
Related topic	ProjectiveSpecialLinearGroup