Orbit Decomposition Lemma

Orbit Decomposition Lemma: Let $G$ be a group acting on a set $X$ via a group action . Then the set of orbits $\{G\cdot x : x\in X\}$ is a partition of $X$.

Equivalently, define a relation $\sim$ on $X$ by $x\sim y$ if there exists $g\in G$ with $g\cdot x=y$. Then $\sim$ is an equivalence relation , and its equivalence classes are exactly the orbits.

Proof sketch: Reflexivity uses the identity element: $e\cdot x=x$. Symmetry uses inverses: if $g\cdot x=y$ then $g^{-1}\cdot y=x$. Transitivity uses multiplication: if $g\cdot x=y$ and $h\cdot y=z$ then $(hg)\cdot x=z$. Hence orbits are equivalence classes and therefore form a partition.