# Category:Examples of Group Actions

Jump to navigation
Jump to search

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.

### C

- Conjugacy Action (12 P)

### G

- Group Action on Coset Space (6 P)

### S

- Subgroup Action (3 P)
- Subset Product Action (4 P)

### T

- Trivial Group Action (1 P)

## Pages in category "Examples of Group Actions"

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