This may be a straightforward question, but if I have a group $G$ acting on a set $A$, and two elements $a,b\in A$ belong to the same orbit, how do I show that their stabilizers are conjugate.
So far I know that $a=gb$ for some $g\in G$. Do I just need to show that $g$ times some element of the stabilizer of $b$ is equal to a stabilizer of $a$?
The stabilizer $A$ of $a$ is the $g$ conjugate of the stabilizer $B$ of $b$ : $$h\in A \iff ha= a \iff hg^{-1}bg=g^{-1}bg \iff ghg^{-1}\in B$$ In other words, $B=gAg^{-1}$