# 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 $\struct {G, \circ}$ be a group whose identity is $e$.

### Left Group Action

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:

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

### Right Group Action

A right group action is a mapping $\phi: X \times G \to X$ such that:

$\forall \tuple {x, g} \in X \times G : x * g := \map \phi {x, g} \in X$

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

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

The group $G$ thus acts on the set $X$.

The group $G$ can be referred to as the group of transformations, or a transformation group.

## Subcategories

This category has the following 6 subcategories, out of 6 total.

## Pages in category "Group Actions"

The following 32 pages are in this category, out of 32 total.