Group actions induce an equivalence relation over a set's orbits

up:: Orbit of Group Action

Let there be a Group Action of $G$ over $X$, and let $x\in X$.

Define the Equivalence Relation over $X$ as

$$x\sim y⟺\exists g\in G:y=g\cdot x$$

Note that it is an equivalence relation, since

• $x\sim y$ via $g\in G$ $⟺$ $y\sim x$ via $g^{-1}\in G$
• $x\sim y$ via $g$, and $y\sim z$ via $h$ $⟹$ $x\sim z$ via $gh$
• $x\sim x$ via $e_G$

Note that the Quotient Space is the set of Orbits¹, since

$$Or{b}_x=\{y\in X\mid \exists g\in G:y=g\cdot x\}=[x]$$

Thus, we have that a group action partitions a set into its orbits (under this action).

Footnotes

1. I.e. orbits $⟺$ equivalence classes. ↩