The question at hand:
The group $GL_2(\mathbb F_p)$ of all $2\times 2$ invertible matrices with coefficients in the finite field $\mathbb F_p$ has order $(p^2 − 1)(p^2 − p)$. Let $SL_2(\mathbb F_p)$ be the subgroup consisting of all matrices of determinant $1$, i.e. $SL_2(\mathbb F_p)$ is the kernel of the group homomorphism $\det : GL_2(\mathbb F_p) → \mathbb F_p$. Compute the order of $SL_2(\mathbb F_p)$. Hint: $GL_2(\mathbb F_p)$ is a disjoint union of fibers of $\det$.
I am feeling a little lost on this problem. I understand via my professor that $GL_2(F_p)$ is the union of the fibres of det, that each fibre is a coset of $SL_2(F_p)$, and that $GL_2(F_p)$ is the order of $SL_2(F_p)$ times the number of fibres. However, I am not sure how to show that the order is $(p^2-1)(p^2-p)$. Does anyone have any insight into this problem?