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 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:

 A: 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.
A: First of all, Katok's treatment of proper discontinuity is partially erroneous, see the discussion and examples here. 
Secondly, when she says that an orbit is "discrete", it is not immediate that the orbit is also closed, which is what one needs for local finiteness of the orbit. (As an example, the subset $\{n^{-1}: n\in {\mathbb N}\}$ of ${\mathbb R}$ is discrete but is not locally finite.) However, what she is really interested in is isometric actions. Under this assumption, one has:
Theorem. Suppose that $G$ is a (sub)group of isometries  of a metric space $X$. Then the following are equivalent:


*

*The action of $G$ on $X$ is properly discontinuous in the usual sense (for every compact subset $K\subset X$ the subset $\{g\in G: gK\cap K\ne \emptyset\}$ is finite).  

*For every $x\in X$ the stabilizer $G_x$ is finite and the orbit $Gx$ is a discrete subset of $X$ (i.e. every orbit point is isolated in $Gx$). 

*For every $x\in X$ the orbit map $o_x: g\mapsto gx$ is a proper map $G\to X$, where $G$ is equipped with discrete topology, i.e. preimages of compact subsets of $X$ under $o_x$ are finite. (This is a clean way to say that "$G$-orbits in $X$ are locally finite.") 

*In addition, if $X$ is a Heine-Borel space (i.e. closed and bounded subsets of $X$ are compact), which is the only case Katok cares about, in 2 and 3 one can replace "every" with "some". 
