Determine the orders \(|\operatorname{GL}_n(\mathbb{F}_p)|\) and \(|\operatorname{SL}_n(\mathbb{F}_p)|\) for any prime \(p\) and integer \(n \ge 1\).
Solution
An invertible \(n\times n\) matrix over \(\mathbb F_p\) is exactly an ordered basis of \(\mathbb F_p^n\) written as columns. The first column has \(p^n-1\) choices, the second has \(p^n-p\) choices, and in general the \((k+1)\)-st column has \(p^n-p^k\) choices. Hence \[ \boxed{|\operatorname{GL}_n(\mathbb F_p)|=\prod_{k=0}^{n-1}(p^n-p^k).} \] Equivalently, \[ |\operatorname{GL}_n(\mathbb F_p)| =p^{n(n-1)/2}\prod_{j=1}^n(p^j-1). \]
The determinant map \[ \det:\operatorname{GL}_n(\mathbb F_p)\to\mathbb F_p^\times \] is surjective, because \[ \det\operatorname{diag}(a,1,\dots,1)=a \] for every \(a\in\mathbb F_p^\times\). Its kernel is \(\operatorname{SL}_n(\mathbb F_p)\). Therefore \[ \boxed{|\operatorname{SL}_n(\mathbb F_p)|= \frac1{p-1}\prod_{k=0}^{n-1}(p^n-p^k).} \]