Conjugacy is Equivalence Relation

Theorem
Conjugacy of group elements is an equivalence relation.

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

Reflexive

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

Thus conjugacy of group elements is reflexive.

Symmetric
Thus conjugacy of group elements is symmetric.

Transitive
Thus conjugacy of group elements is transitive.

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