It's easy to see that an immediate corollary of orbit-stabilizer is that the orbit of any element under the action of a finite $p$-group has cardinality a power of $p$ (this question is not a duplicate, as the related questions only show the result for finite groups).
Naturally, I've tried to generalize the idea to infinite groups, but to no avail. That is to say, if $G$ is an infinite
$p$-group which acts on a set $X$, such that the orbit of some $x \in X$ is finite, we would like to show that the size of said orbit is a power of $p$.
Some ideas so far: We know that the size of the orbit of $x$ is equal to $[G : G_x]$, where $G_x$ is the stabilizer of $x$ (by orbit stabilizer). Then, if it were the case that $[G : G_x]$ had a different prime factor $q \neq p$, we could write $q \mid |G / G_x|$. However, we can't exactly apply Cauchy's theorem here, because $G / G_x$ is not necessarily a group. My next idea was to try to find some nontrivial quotient group on which we could apply Cauchy's theorem, but I haven't been successful thus far. Any help would be appreciated.
Best Answer
Since $G_x$ is a subgroup of finite index in $G$ there is a normal subgroup $N$ contained in $G_x$ which also has finite index in $G$. (Well-known result I believe.)
I think that's all you need.