Exercise 1705Algebra › GroupsM1Added 3 Nov 2026

The order of $GL_n(\mathbb{F}_p)$

#general linear group#finite field#order

Statement

Compute $|GL_2(\mathbb{F}_p)|$ for $p$ prime.

Hint

Count invertible $2\times2$ matrices over $\mathbb{F}_p$ by choosing columns.

Show solutionHide solution

First column. A nonzero vector in $\mathbb{F}_p^2$ : $p^2-1$ choices.

Second column. Must not be in the span of the first: $p^2-p$ choices.

Total. $|GL_2(\mathbb{F}_p)|=(p^2-1)(p^2-p)$ .

Example $p=2$ . $|GL_2(\mathbb{F}_2)|=3\cdot2=6\cong S_3$ .

Conclusion. $|GL_2(\mathbb{F}_p)|=(p^2-1)(p^2-p)=p(p-1)^2(p+1)$ .