\begin{definition}[Definition:Group Action/Left Group Action]
Let $X$ be a set.
Let $\struct {G, \circ}$ be a group whose identity is $e$.
A '''(left) group action''' is an operation $\phi: G \times X \to X$ such that:
:$\forall \tuple {g, x} \in G \times X: g * x := \map \phi {g, x} \in X$
in such a way that the group action axioms are satisfied:
{{:Definition:Group Action Axioms}}
\end{definition}