Orbits partition the set

Statement

Let a group $G$ act on a set $X$. The relation $x \sim y$ if and only if $y = g \cdot x$ for some $g \in G$ is an equivalence relation on $X$. Its equivalence classes are the orbits, so each point of $X$ lies in exactly one orbit.

Remarks

Reflexivity uses $e \cdot x = x$, symmetry uses $x = g^{-1} \cdot (g \cdot x)$, and transitivity uses $h \cdot (g \cdot x) = (hg) \cdot x$. If $X$ is finite, then $|X|$ is the sum of the sizes of the distinct orbits. The cosets of a subgroup and the conjugacy classes of a group are examples of orbits.

Explanations

L1 human

The group action cuts the set into separate pieces, the orbits. Inside one piece, each point can reach each other point. No point can reach a point of another piece.