# Definition:Conjugate (Group Theory)/Element

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## Contents

## Definition

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:

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

This can be voiced as:

**$x$ is the conjugate of $y$ (by $a$ in $G$)**

or:

**$x$ is conjugate to $y$ (by $a$ in $G$)**

## Also defined as

Some sources define the **conjugate of $x$ by $a$ in $G$** as:

- $x \sim y \iff \exists a \in G: x \circ a = a \circ y$

or:

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

## Also known as

Some sources refer to the **conjugate** of $x$ as the **transform** of $x$.

Some sources refer to **conjugacy** as **conjugation**.

## Also see

- Results about
**conjugacy**can be found here.