In chapter 9 of the book "New directions in locally compact groups" the cardinal of the set of products $xy$ in $\text{SL}_2(F_p)$ with $x$ is lower triangular and $y$ is upper triangular was written to be $(p-1)p^2$. Can anyone explain how this calculation is done?
Here, $F_p$ is the field with $p$ elements, where $p$ is a prime number and $\text{SL}_2(F_p)$ is the set of $2\times2$ matrices with determinant equals to 1.
There are $(p-1)p$ choices for $x$ (choose upper left and lower left entry, everthing else is forced) and $(p-1)p$ choices for $y$. The product $xy$ and $x'y'$ are the same if and only if there is a diagonal matrix $M$ with $x = x'M$ and $y = M^{-1}y'$, and there are $(p-1)$ such diagonal matrices.
It follows that there are a total of $(p-1)^2p^2/(p-1) = (p-1)p^2$ possible matrices that can be written in this form.