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**vincisonfire** $\displaystyle |\text{GL}_2(\mathbb{F})/\text{SL}_2(\mathbb{F})| = |\mathbb{F}^{\times}| = q-1 $ because any element except 0 has a multiplicative inverse in $\displaystyle \mathbb F $.

In other word, there are $\displaystyle q-1 $ equivalent classes (or orbits) of $\displaystyle \text{SL}_2(\mathbb{F}) $.

We thus need to find the number of elements in each orbit.

In an equivalent class we have

$\displaystyle A=\left( \begin{array}{cc}q&a\\0&q^{-1} \end{array} \right) $ $\displaystyle A \in \text{SL}_2(\mathbb{F}) $

There are $\displaystyle \frac{q}{2} $combinations of $\displaystyle q \text{ and } q^{-1} $.

$\displaystyle 2q $combinations of $\displaystyle a $because we can swap $\displaystyle a \text{ and } 0 $.

There are therefore $\displaystyle q^2 $elements in its equivalent classes.

$\displaystyle |\text{SL}_2(\mathbb{F})| = q^2(q-1) = q^3-q $

Am I right?