Stabilizer subgroups - proof verification

I have a problem that I would like help on. I'm preparing for an exam, and I have provided my work below.

Problem statement: Let G act on X, and suppose x,y in X are in the same orbit for this action. How are the stabilizer subgroups G_x and G_y related?

My attempt:G_x = G_y = {g in G: gx=x} = {g in G: gy=y} for all x,y in X, for all g in G.

Thanks.

Re: Stabilizer subgroups - proof verification

The answer is $\displaystyle G_y = hG_x h^{-1} $ for some $\displaystyle h\in G$.

See if you can derive it.