# 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.