Left coset space of centralizer is in bijective correspondence with conjugacy class
Note that this holds for finite groups as well as for infinite groups where the orders are interpreted as (possibly infinite) cardinals.
Consider the action of on itself by conjugation (by fact (1)). By fact (2), we can identify the orbit of the point in the set with the left coset space of the stabilizer of in , which is the subgroup . This completes the proof.