# 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 $e$.

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

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

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

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

\((GA \,2)\) | $:$ | \(\displaystyle \forall x \in X:\) | \(\displaystyle e * x = x \) |

## Subcategories

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

### E

### O

### S

### T

## Pages in category "Group Actions"

The following 35 pages are in this category, out of 35 total.

### C

### G

- Group Action defines Permutation Representation
- Group Action determines Bijection
- Group Action Induces Equivalence Relation
- Group Action on Prime Power Order Subset
- Group Action on Sets with k Elements
- 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