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:

\(\ds g_1 * \paren {g_2 * s}\)

\(=\)

\(\ds g_1 * s\)

Definition of $*$

\(\ds \)

\(=\)

\(\ds s\)

Definition of $*$

\(\ds \)

\(=\)

\(\ds \paren {g_1 \circ g_2} * s\)

Definition of $*$

demonstrating that Group Action Axiom $\text {GA} 1$ holds.

Then:

\(\ds e * s\)

\(=\)

\(\ds s\)

Definition of $*$

demonstrating that Group Action Axiom $\text {GA} 2$ holds.

$\blacksquare$