Equivalence of Definitions of Saturation Under Group Action

Definition

Let $G$ be a group acting on a set $X$.

Let $S\subset X$ be a subset.

The following definitions of its saturation are equivalent:

Definition 1

The saturation of $S$ is its saturation by the equivalence relation induced by the action.

Definition 2

The saturation of $S$ is the union of its images under the group action:

$\overline S = \ds \bigcup_{g \mathop \in G} g S$

Proof

Let $x\in X$.

We have:

\(\ds x\)

\(\in\)

\(\ds \bigcup_{g \mathop \in G} g S\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \exists g \in G: \, \)

\(\ds x\)

\(\in\)

\(\ds g S\)

Definition of Set Union

\(\ds \leadstoandfrom \ \ \)

\(\ds \exists g \in G : \exists s \in S: \, \)

\(\ds x\)

\(=\)

\(\ds g s\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \exists s \in S : \exists g \in G: \, \)

\(\ds x\)

\(=\)

\(\ds g s\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \exists s \in S: \, \)

\(\ds s\)

\(\RR_G\)

\(\ds x\)

Definition of Equivalence Relation Induced by Group Action

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(\in\)

\(\ds \overline S\)

Definition of Saturation Under Equivalence Relation

$\blacksquare$