i am new in this forum.
My question is about group actions
We have a transitive action of $G$ and $\beta$ a element in the fixed points of the stabilizer of another element $\alpha$. Then $\alpha$ and $\beta$ have the same stabilizers?
Thank you for the further answers
If it's finite groups you are talking about, then the following argument applies.
Your hypothesis is that $G_{\alpha} \le G_{\beta}$. Since $G$ acts transitively, there is $g \in G$ such that $\alpha g = \beta$, and then $g^{-1} G_{\alpha} g = G_{\beta}$.
If $G$ is finite, this implies that $G_{\alpha}$ and $G_{\beta}$ have the same order, so they are equal.
If $G$ is infinite, though, this isn't necessarily true. There are examples of groups $G$ with a subgroup $H$ and an element $g \in G$ such that $H$ is properly contained in $g^{-1} H g$. If you let $G$ act by right multiplication on the cosets $H a$, and take $\alpha = H$, $\beta = H g$, then $G_{\alpha} = H$ is properly contained in $G_{\beta} = g^{-1} G_{\alpha} g = g^{-1} H g$.