# Category:Group Actions

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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:

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

\((\text {GA} 2)\) | $:$ | \(\ds \forall x \in X:\) | \(\ds e * x = x \) |

### 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 6 subcategories, out of 6 total.

### E

### O

- Orbit-Stabilizer Theorem (4 P)

### S

- Stabilizer is Subgroup (2 P)

### T

- Topological Group Actions (2 P)
- Transitive Group Actions (7 P)

## Pages in category "Group Actions"

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

### C

### G

- Group Action defines Permutation Representation
- Group Action determines Bijection
- Group Action Induces Equivalence Relation
- Group Action on Subgroup by Left Regular Representation
- Group Action on Subgroup by Right Regular Representation
- Group Action on Subgroup of Symmetric Group
- Group Acts by Homeomorphisms Implies Projection on Quotient Space is Open
- Group Acts on Itself