Order of general- and special linear groups over finite fields. Let $\mathbb{F}_3$ be the field with three elements. Let $n\geq 1$. How many elements do the following groups have?


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*$\text{GL}_n(\mathbb{F}_3)$

*$\text{SL}_n(\mathbb{F}_3)$


Here GL is the general linear group, the group of invertible n×n matrices, and SL is the special linear group, the group of n×n matrices with determinant 1. 

Determinant function is a surjective homomorphism from $GL(n, F)$ to $F^*$ with kernel $SL(n, F)$. Hence by the fundamental isomorphism theorem $\frac{GL(n,F)}{SL(n,F)}$ is isomorphic to $F^*$, the multiplicative group of nonzero elements of $F$. 
Thus if $F$ is finite with $p$ elements then $|GL(n,F)|=(p-1)|SL(n, F)|$.
