\begin{definition}[Definition:Group Action Axioms]
Let $\struct {G, \circ}$ be a group which acts on a set $X$.
The properties that define the group action $*: G \times X \to X$ are summarised as:
{{begin-axiom}}
{{axiom | n = \text {GA} 1
        | q = \forall g, h \in G, x \in X
        | m = g * \paren {h * x} = \paren {g \circ h} * x
}}
{{axiom | n = \text {GA} 2
        | q = \forall x \in X
        | m = e * x = x
}}
{{end-axiom}}
These properties can be referred to as the '''group action axioms'''.
Category:Definitions/Group Actions
\end{definition}