Stabilisers of points in the same orbit are conjugate

Statement

Let a group $G$ act on a set $X$, let $x \in X$, and let $g \in G$. Then the stabiliser of $g \cdot x$ is $G_{g \cdot x} = g G_x g^{-1}$.

Remarks

An element $h$ fixes $g \cdot x$ if and only if $g^{-1}hg$ fixes $x$. So the stabilisers of the points of one orbit are conjugate subgroups. In particular, they all have the same order.

Explanations

L1 human

If $g$ moves $x$ to $y$, then the elements that fix $y$ are the elements that fix $x$, seen through $g$. To fix $y$, you can move $y$ back to $x$, fix $x$, and move $x$ to $y$ again. So the two stabilisers are conjugate.