TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-03

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The orbits of a group action are the equivalence classes of $x\sim y$ iff $y=g\cdot x$ for some $g$ , and hence partition the acted-on set

Statement

For a left action of $G$ on $X$ , define $x\sim y$ when $y=g\cdot x$ for some $g\in G$ . This is an equivalence relation, its equivalence class at $x$ is $G\cdot x$ , and the distinct orbits partition $X$ .

Facts & Assumptions

Given: A left action of a group $G$ on a set $X$ .

[L1]

The action laws are $e\cdot x=x$ and $(gh)\cdot x=g\cdot(h\cdot x)$ (Left group actions, transitive actions, and faithful actions).

[L2]

The orbit at $x$ is $G\cdot x=\{g\cdot x:g\in G\}$ (The orbit $G\cdot x$ and stabilizer $G_x$ of a point in a group action).

[L3]

Equivalence classes of an equivalence relation partition the underlying set (Equivalence relation, equivalence class, and the quotient set $A/{\sim}$ , The equivalence classes of an equivalence relation are nonempty, cover $A$ , and are pairwise equal or disjoint; conversely every such cover arises from exactly one equivalence relation).

Proof

technique · direct

1.1

The relation is reflexive: $x=e\cdot x$ , so $x\sim x$ .

L1given

1.2

If $y=g\cdot x$ , then $x=g^{-1}\cdot y$ , so $x\sim y$ implies $y\sim x$ .

L1givenalgebra

1.3

If $y=g\cdot x$ and $z=h\cdot y$ , then $z=(hg)\cdot x$ , so $x\sim y$ and $y\sim z$ imply $x\sim z$ .

L1givenalgebra

2.1

Steps 1.1–1.3 show that $\sim$ is an equivalence relation. Its class at $x$ is precisely the set of $y=g\cdot x$ , namely $G\cdot x$ .

step 1.1step 1.2step 1.3L2L3

3.1

Therefore the distinct orbits partition $X$ .

step 2.1L3∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 33 results over 17 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Brosnan, Orbits and stabilizers (standard reference, not scraped)

The orbits of a group action are the equivalence classes of x∼yx\sim yx∼y iff y=g⋅xy=g\cdot xy=g⋅x for some ggg, and hence partition the acted-on set