Cartesian Product of Group Actions

Theorem
Let $\struct {G, \circ}$ be a group.

Let $S$ and $T$ be sets.

Let $*_S: G \times S \to S$ and $*_T: G \times T \to T$ be group actions.

Then the operation $*: G \times \paren {S \times T} \to S \times T$ defined as:
 * $\forall \tuple {g, \tuple {s, t} } \in G \times \paren {S \times T}: g * \tuple {s, t} = \tuple {g *_S s, g *_T t}$

is a group action.

Proof
The group action axioms are investigated in turn.

Let $g, h \in G$ and $s, t \in S$.

Thus:

demonstrating that group action axiom $GA\,1$ holds.

Then:

demonstrating that group action axiom $GA\,2$ holds.

The group action axioms are thus seen to be fulfilled, and so $*$ is a group action.