# Category:Conjugacy

This category contains results about Conjugacy in the context of Group Theory.

Definitions specific to this category can be found in Definitions/Conjugacy.

Let $\struct {G, \circ}$ be a group.

### Definition 1

The **conjugacy relation** $\sim$ is defined on $G$ as:

- $\forall \tuple {x, y} \in G \times G: x \sim y \iff \exists a \in G: a \circ x = y \circ a$

### Definition 2

The **conjugacy relation** $\sim$ is defined on $G$ as:

- $\forall \tuple {x, y} \in G \times G: x \sim y \iff \exists a \in G: a \circ x \circ a^{-1} = y$

## Subcategories

This category has the following 7 subcategories, out of 7 total.

### C

### I

### S

## Pages in category "Conjugacy"

The following 31 pages are in this category, out of 31 total.

### C

- Conjugacy Class Equation
- Conjugacy is Equivalence Relation
- Conjugate of Commuting Elements
- Conjugate of Set by Group Product
- Conjugate of Set by Identity
- Conjugate of Set with Inverse Closed for Inverses
- Conjugate of Set with Inverse is Closed
- Conjugate of Subgroup is Subgroup
- Conjugate Permutations have Same Cycle Type
- Conjugates of Elements in Centralizer
- Cycle Decomposition of Conjugate

### E

### N

### O

### S

- Subgroup equals Conjugate iff Normal
- Subgroup is Normal iff Contains Conjugate Elements
- Subgroup is Normal iff Left Coset Space is Right Coset Space
- Subgroup is Normal iff Left Cosets are Right Cosets
- Subgroup is Normal iff Normal Subset
- Subgroup is Subset of Conjugate iff Normal
- Subgroup is Superset of Conjugate iff Normal
- Subset has 2 Conjugates then Normal Subgroup