This is simple and may have been asked before, but I couldn't find it.
I have been asked to 'Define the action of $SL_2(p)$ (the group of 2 by 2 matrices of determinant 1 with entries in $\mathbb{F_p}$, the integers mod some prime p) on $\mathbb{F_p} \cup {\infty}$ by Möbius transformations.
It seems to me that if one does this in the natural way, with $\frac{ax+b}{cx+d}$ for some $x\in\mathbb{F_p} \cup {\infty}$, then a lot of the time your result will no longer be in this set. Is this wrong, and how would you define the action if this is incorrect?