Trivial Group Action is Group Action

Theorem
Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $S$ be a set.

Let $*: G \times S \to S$ be the trivial group action:
 * $\forall \tuple {g, s} \in G \times S: g * s = s$

Then $*$ is indeed a group action.

Proof
The group action axioms are investigated in turn.

Let $g_1, g_2 \in G$ and $s \in S$.

Thus:

demonstrating that holds.

Then:

demonstrating that holds.