# Definition:Group Action Axioms

## 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**.