# Category:Group Actions

This category contains results about Group Actions.
Definitions specific to this category can be found in Definitions/Group Actions.

Let $X$ be a set.

Let $\left({G, \circ}\right)$ be a group whose identity is $e$.

A (left) group action is an operation $\phi: G \times X \to X$ such that:

$\forall \left({g, x}\right) \in G \times X: g * x := \phi \left({\left({g, x}\right)}\right) \in X$

in such a way that the group action axioms are satisfied:

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