# Category:Examples of Group Actions

This category contains examples of Group Action.

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:

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

### 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:

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

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 9 subcategories, out of 9 total.

## Pages in category "Examples of Group Actions"

The following 3 pages are in this category, out of 3 total.