> [!NOTE] Theorem (Orbits Under Group Action are Equivalence Classes) > Let $G$ [[Group action|act]] on $X$. For $x,y\in X$ write $x\sim y$ if, and only if there is some $g\in G$ such that $g*x=y$ (i.e. $y$ lies in the [[Orbit under Group Action|orbit]] of $x$ under $G$). Then $\sim$ is an [[Equivalence relations|equivalence relation]]. ###### Proof By definition of group actions, $\text{Id}*x=x$ and so $x\sim x$ i.e. $\sim$ is reflexive. If $x\sim y$ then $g*x=y$ for some $g\in G$ which yields $x=g^{-1}*y$ since group elements have inverses, and so $\sim$ is symmetric. Finally suppose $x\sim y$ and $y \sim z$. Then there exists $g,h\in G$ such that $g*x=y$ and $h*y=z$. By definition of group actions, $(hg)*x=h*(g*x)=h*y=z$ and so $x\sim z$. i.e. $\sim$ is transitive.