Group Action Induces Equivalence Relation
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Theorem
Let $G$ be a group whose identity is $e$.
Let $X$ be a set.
Let $*: G \times S \to S$ be a group action.
Let $\RR_G$ be the relation induced by $G$, that is:
- $x \mathrel {\RR_G} y \iff y \in \Orb x$
where:
- $\Orb x$ denotes the orbit of $x \in X$.
Then:
- $\RR_G$ is an equivalence relation.
- The equivalence class of an element is its orbit.
Proof
Let $x \mathrel {\RR_G} y \iff y \in \Orb x$.
Checking in turn each of the criteria for equivalence:
Reflexivity
$x = e * x \implies x \in \Orb x$ from the definition of group action.
Thus $\RR_G$ is reflexive.
$\Box$
Symmetry
\(\ds y\) | \(\in\) | \(\ds \Orb x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \exists g \in G: \, \) | \(\ds y\) | \(=\) | \(\ds g * x\) | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds g^{-1} * \paren {g * x}\) | \(=\) | \(\ds g^{-1} * y\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds g^{-1} * y\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \exists g^{-1} \in G: \, \) | \(\ds x\) | \(=\) | \(\ds g^{-1} * y\) | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(\in\) | \(\ds \Orb y\) |
Thus $\RR_G$ is symmetric.
$\Box$
Transitivity
\(\ds y\) | \(\in\) | \(\ds \Orb x\) | ||||||||||||
\(\ds \land \ \ \) | \(\ds z\) | \(\in\) | \(\ds \Orb y\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \exists g_1 \in G: \, \) | \(\ds y\) | \(=\) | \(\ds g_1 * x\) | ||||||||||
\(\ds \land \ \ \) | \(\ds \exists g_2 \in G: \, \) | \(\ds z\) | \(=\) | \(\ds g_2 * y\) | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds z\) | \(=\) | \(\ds g_2 * \paren {g_1 * x}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds z\) | \(=\) | \(\ds \paren {g_2 g_1} * x\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds z\) | \(\in\) | \(\ds \Orb x\) |
Thus $\RR_G$ is transitive.
$\Box$
So $\RR_G$ has been shown to be an equivalence relation.
Hence the result, by definition of an equivalence class.
$\blacksquare$
Also see
- Definition:Set of Orbits
- Definition:Equivalence Relation Induced by Group Action
- Set of Orbits forms Partition
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.5$. Orbits: Lemma $\text{(i)}$
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.6$: Exercise $4.2$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Sylow Theorems: $\S 54$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.3$: Group actions and coset decompositions
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Proposition $10.13$