# Category:Group Actions

From ProofWiki

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 $\left({G, \circ}\right)$ be a group whose identity is $e$.

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

- $\forall \left({g, x}\right) \in G \times X: g * x := \phi \left({\left({g, x}\right)}\right) \in X$

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

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

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

## Subcategories

This category has the following 4 subcategories, out of 4 total.

### E

### O

### S

### T

## Pages in category "Group Actions"

The following 61 pages are in this category, out of 61 total.

### C

- Cartesian Product of Group Actions
- Center of Group is Kernel of Conjugacy Action
- Conjugacy Action is Group Action
- Conjugacy Action is not Transitive
- Conjugacy Action on Group Elements is Group Action
- Conjugacy Action on Identity
- Conjugacy Action on Subgroups is Group Action
- Conjugation Action on Abelian Group is Trivial
- Correspondence Between Group Actions and Permutation Representations
- Correspondence between Linear Group Actions and Linear Representations

### G

- Group Action defines Permutation Representation
- Group Action determines Bijection
- Group Action Induces Equivalence Relation
- Group Action of Symmetric Group
- Group Action of Symmetric Group Acts Transitively
- Group Action on Coset Space
- 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 by Right Regular Representation is not Transitive
- Group Action on Subgroup of Symmetric Group
- Group Action on Subset of Group
- Group Acts by Homeomorphisms Implies Projection on Quotient Space is Open
- Group Acts Effectively on Left Coset Space
- Group Acts on Itself

### L

### O

- Orbit of Conjugacy Action on Subgroup is Set of Conjugate Subgroups
- Orbit of Element of Group Acting on Itself is Group
- Orbit of Element under Conjugacy Action is Conjugacy Class
- Orbit of Group Action on Subgroup by Right Regular Representation is Right Coset
- Orbit of Subgroup under Coset Action is Coset Space
- Orbit of Trivial Group Action is Singleton
- Orbit-Stabilizer Theorem
- Orbits of Group Action on Sets with Power of Prime Size

### P

### R

### S

- Set of Orbits is Partition
- Set of Permutations is Largest Effective Transformation Group
- Stabilizer in Group of Transformations
- Stabilizer is Subgroup
- Stabilizer is Subgroup/Corollary 1
- Stabilizer is Subgroup/Corollary 2
- Stabilizer of Cartesian Product of Group Actions
- Stabilizer of Conjugacy Action on Subgroup is Normalizer
- Stabilizer of Coset Action on Power Set
- Stabilizer of Element of Group Acting on Itself is Trivial
- Stabilizer of Element under Conjugacy Action is Centralizer
- Stabilizer of Subset Product Action on Power Set
- Stabilizers in Group Action on Subset