Left coset space of centralizer is in bijective correspondence with conjugacy class

Statement
For a group $$G$$ and an element $$x$$ in $$G$$, there is a bijection between the space of fact about::left cosets of the fact about::centralizer $$C_G(x)$$ in $$G$$ (denoted $$G/C_G(x)$$) and the fact about::conjugacy class $$c$$ of $$g$$ in $$G$$.

In particular:

$$|c| = [G:C_G(x)]$$

Note that this holds for finite groups as well as for infinite groups where the orders are interpreted as (possibly infinite) cardinals.

Applications

 * Size of conjugacy class divides index of center
 * Size of conjugacy class divides order of group

Other related facts/indirect applications

 * Class equation of a group

Facts used

 * 1) uses::Group acts as automorphisms by conjugation
 * 2) uses::Fundamental theorem of group actions

Proof outline
Consider the action of $$G$$ on itself by conjugation (by fact (1)). By fact (2), we can identify the orbit of the point $$x$$ in the set $$G$$ with the left coset space of the stabilizer of $$x$$ in $$G$$, which is the subgroup $$C_G(x)$$. This completes the proof.