Definition:Group Action Axioms

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Definition

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:

\((GA \, 1)\)   $:$     \(\displaystyle \forall g, h \in G, x \in X:\) \(\displaystyle g * \paren {h * x} = \paren {g \circ h} * x \)             
\((GA \, 2)\)   $:$     \(\displaystyle \forall x \in X:\) \(\displaystyle e * x = x \)             


These properties can be referred to as the group action axioms.