# Definition:Conjugate (Group Theory)/Element

## 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.