Orbit of Trivial Group Action is Singleton

Theorem

Let $\left({G, \circ}\right)$ be a group whose identity is $e$.

Let $S$ be a set.

Let $*: G \times S \to S$ be the trivial group action:

$\forall \left({g, s}\right) \in G \times S: g * s = s$

Let $s \in S$.

Then the orbit of $s$ under $*$ is $\left\{{s}\right\}$.

Proof

By definition:

$\operatorname{Orb} \left({s}\right) = \left\{{t \in S: \exists g \in G: g * s = t}\right\}$

By definition of the trivial group action:

$\forall g \in G: g * s = s$

Hence the result.

$\blacksquare$

Sources

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.3$: Group actions and coset decompositions: Examples of group actions: $\text{(iii)}$