# Conjugacy is Equivalence Relation

## Proof

Checking each of the criteria for an equivalence relation in turn:

### Reflexive

$\forall x \in G: e_G \circ x = x \circ e_G \implies x \sim x$

$\Box$

### Symmetric

 $\displaystyle$  $\displaystyle x \sim y$ $\displaystyle$ $\leadsto$ $\displaystyle a \circ x = y \circ a$ Definition of Conjugate of Group Element $\displaystyle$ $\leadsto$ $\displaystyle a \circ x \circ a^{-1} = y$ Definition of Group $\displaystyle$ $\leadsto$ $\displaystyle a^{-1} \circ y = x \circ a^{-1}$ Definition of Group $\displaystyle$ $\leadsto$ $\displaystyle y \sim x$ Definition of Conjugate of Group Element

$\Box$

### Transitive

 $\displaystyle$  $\displaystyle x \sim y, y \sim z$ $\displaystyle$ $\leadsto$ $\displaystyle a_1 \circ x = y \circ a_1, a_2 \circ y = z \circ a_2$ Definition of Conjugate of Group Element $\displaystyle$ $\leadsto$ $\displaystyle a_2 \circ a_1 \circ x = a_2 \circ y \circ a_1 = z \circ a_2 \circ a_1$ Definition of Group $\displaystyle$ $\leadsto$ $\displaystyle x \sim z$ Definition of Conjugate of Group Element

$\Box$

All criteria are satisfied, and so conjugacy of group elements is shown to be an equivalence relation.

$\blacksquare$