# Equivalence of Definitions of Conjugate of Group Element

## Theorem

The following definitions of the concept of Conjugate of Group Element are equivalent:

### 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:

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

### 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:

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

## Proof

 $\ds a \circ x$ $=$ $\ds y \circ a$ $\ds \leadstoandfrom \ \$ $\ds a \circ x \circ a^{-1}$ $=$ $\ds y$

 $\ds x \circ a$ $=$ $\ds a \circ y$ $\ds \leadstoandfrom \ \$ $\ds a^{-1} \circ x \circ a$ $=$ $\ds y$

 $\ds \exists b \in G: \,$ $\ds x \circ b$ $=$ $\ds b \circ y$ $\ds \leadstoandfrom \ \$ $\ds b^{-1} \circ x \circ b \circ b^{-1}$ $=$ $\ds b^{-1} \circ b \circ y \circ b^{-1}$ $\ds \leadstoandfrom \ \$ $\ds b^{-1} \circ x$ $=$ $\ds y \circ b^{-1}$ $\ds \leadstoandfrom \ \$ $\ds \exists a \in G: \,$ $\ds a \circ x$ $=$ $\ds y \circ a$ setting $a := b^{-1}$

$\blacksquare$