Stabilizers: If $X$ is a $G$-set and $x, y \in X$ such that $y=hx$ for some $h\in G$, show that $G_x$ and $G_y$ are isomorphic

Question

If $X$ is a $G$-set and $x, y \in X$ such that $y=hx$ for some $h\in G$, we need to show that $G_x$ and $G_y$ are isomorphic (where $G_x$ represents the stabilizer of $x$ in $G$).

I was able to prove that $G_y = hG_x h^{-1}$. Can I use this to prove the isomorphism between $G_x$ and $G_y$ or it proves the bijection?

Answer 1

Essentially, yes.

Conjunction by $h$ is an isomorphism from $G_x$ to $G_y$. Its inverse is conjugation by $h^{-1}$.

Showing that it is a homomorphism is routine. Indeed, for $s,t\in G_x$, we have

$$\begin{align} h(st)h^{-1}&=hseth^{-1}\\ &=hs(h^{-1}h)th^{-1}\\ &=(hsh^{-1})(hth^{-1}). \end{align}$$

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Recall that $$hG_xh^{-1}=\{ hah^{-1}\in G\mid a\in G_x\}.$$