A finite group $G$ acts on a finite set $X$, the action of $g \in G$ on $x \in X$ being denoted by $gx$. For each $x \in X$ the stabilizer of $x$ is the subgroup $G_x = \{g \in G : gx = x\}$. If $x, y ∈ G$ and if $y = gx$, then express $G_y$ in terms of $G_x$.
How can I able to solve this problem? I am totally stuck on it.
If $y = gx$ then $G_{y} = gG_{x}g^{-1}$ as we can easily verify.
Infact if $h\cdot y = y \Rightarrow h\cdot g\cdot x = g\cdot x \Rightarrow (g^{-1}hg)\cdot x = x \Rightarrow h\in gG_{x}g^{-1} $.
Viceversa if $k \in G_{x}$ then $(gkg^{-1})\cdot y = (gkg^{-1})\cdot (g\cdot x) = g\cdot x = y \Rightarrow gkg^{-1} \in G_{y}$.