Orbit-Stabalizer Theorem

Let $(G,\star )$ : Group. Let $\phi :G\text{texttimes}S\to S$ : Group Action. Let $\sim :S\text{texttimes}S$ be defined as:

$$s\sim t⟺\exists g\in G.gs=t$$

Then this is an equivalence relation:

• Reflexivity: $es=s$
• Symmetry: $gs=t\Rightarrow g^{-}t=s$
• Transitivity: $gs=t,ht=u⟹hgs=u$