In S. Katok's Fuchsian groups, we say that a group G acts properly discontinuously
on $X$ if the G-orbit of any point $x\in X$ is locally finite.
However, I found the assertion "a group acts properly discontinuously on $X$ if and only if each orbit is discrete and the order of the of the stabilizer each point is finite" in the last paragraph in page 27 of this book is not quite correct.
For example, if $X$ is an infinite discrete space and $G=S(X)$ is the group of all bijections of $X$ (all homeomorphisms due to discreteness). Then $G$ acts on $X$ properly discontinuously but the stabilizer each point is actually infinite! (since one can take any bijection that fixes one point).
I found this very frustrating since this equivalence is used many places in this book. Is it possible to add some mild condition to make it hold?
Source:
Best Answer
Note that a point $x$ in the $G$-orbit is not counted just once, it is counted "with a multiplicity equal to the order of $G_x$". In your proposed counterexample, the stabilizer $G_x$ is infinite, as you pointed out. So $x$ is counted with infinite multiplicity, and therefore the orbit $Gx$ is not locally finite.