I'm currently working on a homework question:
Let $G$ be a group. It acts on a set $X$. If $y=a\cdot x$ where $a\in G$, and $\exists g\in G$ s.t. $g\cdot y=y$ and $(a^{-1}ga)\cdot x=x$, how are the stabilizers of $x$ and $y$ related?
I know that when $G$ is abelian, the two stabilizers are identical. But what if the group is non-abelian, are they really related? I can't make any conclusions if this is not abelian.
In general we have $G_{ax} = a G_x a^{-1}$ since $$g \in G_{ax} \iff gax = ax \iff a^{-1} ga x = x \iff a^{-1} g a \in G_x \iff g \in a G_x a^{-1}.$$