# 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

 $\displaystyle a \circ x$ $=$ $\displaystyle y \circ a$ $\displaystyle \iff \ \$ $\displaystyle a \circ x \circ a^{-1}$ $=$ $\displaystyle y$

 $\displaystyle x \circ a$ $=$ $\displaystyle a \circ y$ $\displaystyle \iff \ \$ $\displaystyle a^{-1} \circ x \circ a$ $=$ $\displaystyle y$

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

$\blacksquare$