Enumerating

$$\begin{align*} {\left\lvert {\operatorname{GL}_n({ \mathbf{F} }_p)} \right\rvert} = (p^n-1)(p^n-p)(p^n-p^2)\cdots(p^n - p^{n-1}) .\end{align*}$$

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.