\begin{align*} {\left\lvert {\operatorname{GL}_n({ \mathbf{F} }_p)} \right\rvert} = (p^n-1)(p^n-p)(p^n-p^2)\cdots(p^n - p^{n-1}) .\end{align*}
It suffices to count ordered bases of \({ \mathbf{F} }_p^n\):
- Choose \(\mathbf{v}_1\): there are \(p\) choices for each coefficient, but leave out the vector \(0\), so \(p^n-1\) choices.
- Choose any \(\mathbf{v}_2 \neq \lambda \mathbf{v}_1\), so \(p^n-p\) choices.
- Choose any nonzero \(\mathbf{v}_3 \neq \lambda \mathbf{v}_1 + \eta \mathbf{v}_2\), so \(p^n-p^2\) choices.
- Etc.