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

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