Group Action Induces Equivalence Relation

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Theorem

Let $\struct {G, \circ}$ 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:

$\forall x, y \in X: x \mathrel {\RR_G} y \iff y \in \Orb x$

where:

$\Orb x$ denotes the orbit of $x \in X$.


Then:

$\RR_G$ is an equivalence relation

and:

the equivalence class of an element of $X$ 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

\(\ds \exists e \in G: \, \) \(\ds x\) \(=\) \(\ds e * x\) Group Action Axiom $\text {GA} 2$
\(\ds \leadsto \ \ \) \(\ds x\) \(\in\) \(\ds \Orb x\) Definition of Orbit (Group Theory)

Thus $\RR_G$ is reflexive.

$\Box$


Symmetry

\(\ds y\) \(\in\) \(\ds \Orb x\)
\(\ds \leadsto \ \ \) \(\ds \exists g \in G: \, \) \(\ds y\) \(=\) \(\ds g * x\) Definition of Orbit (Group Theory)
\(\ds \leadsto \ \ \) \(\ds g^{-1} * \paren {g * x}\) \(=\) \(\ds g^{-1} * y\) Group Axiom $\text G 3$: Existence of Inverse Element
\(\ds \leadsto \ \ \) \(\ds \paren {g^{-1} \circ g} * x\) \(=\) \(\ds g^{-1} * y\) Group Action Axiom $\text {GA} 1$
\(\ds \leadsto \ \ \) \(\ds e * x\) \(=\) \(\ds g^{-1} * y\) Group Axiom $\text G 2$: Existence of Identity Element
\(\ds \leadsto \ \ \) \(\ds \exists g^{-1} \in G: \, \) \(\ds x\) \(=\) \(\ds g^{-1} * y\) Group Action Axiom $\text {GA} 2$
\(\ds \leadsto \ \ \) \(\ds x\) \(\in\) \(\ds \Orb y\) Definition of Orbit (Group Theory)

Thus $\RR_G$ is symmetric.

$\Box$


Transitivity

\(\ds y\) \(\in\) \(\ds \Orb x\)
\(\, \ds \land \, \) \(\ds z\) \(\in\) \(\ds \Orb y\)
\(\ds \leadsto \ \ \) \(\ds \exists g_1, g_2 \in G: \, \) \(\ds y\) \(=\) \(\ds g_1 * x\) Definition of Orbit (Group Theory)
\(\, \ds \land \, \) \(\ds z\) \(=\) \(\ds g_2 * y\) Definition of Orbit (Group Theory)
\(\ds \leadsto \ \ \) \(\ds z\) \(=\) \(\ds g_2 * \paren {g_1 * x}\)
\(\ds \leadsto \ \ \) \(\ds \exists g_2 \circ g_1 \in G: \, \) \(\ds z\) \(=\) \(\ds \paren {g_2 \circ g_1} * x\) Group Action Axiom $\text {GA} 1$
\(\ds \leadsto \ \ \) \(\ds z\) \(\in\) \(\ds \Orb x\) Definition of Orbit (Group Theory)

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


Sources

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