Prove that the order of $GL_{2}(\mathbb{{F}_{p}})$ is $p^{4}-p^{3}-p^{2}+p$.

Asked Jan 28 '16 at 12:45

Active Jan 28 '16 at 12:47

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Let $p$ be a prime. Prove that the order of $GL_{2}(\mathbb{{F}_{p}})$ is $p^{4}-p^{3}-p^{2}+p$.

I know that the matrix $2\times 2$ is given by this formula

$$\det\begin{bmatrix}a&b\\c&d\end{bmatrix} = ad-bc.$$

I know that the determinant is going to be zero whenever $ad=bc$.

Thank you.

edited Jan 28 '16 at 12:47

Michael Albanese

asked Jan 28 '16 at 12:45

Smith Pay

Hint How many vectors in $\Bbb F_p^2$ can be the first column of a vector in $GL_2(\Bbb F_p)$? – Travis Willse Jan 28 '16 at 12:47
I believe it should be 2 – Smith Pay Jan 28 '16 at 13:04
Even for $p = 2$, there are three possibilities: $\pmatrix{0\1}$, $\pmatrix{1\0}$, and $\pmatrix{1\1}$. – Travis Willse Jan 28 '16 at 14:11

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