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 left cosets of the centralizer $C_G(x)$ in $G$ (denoted $G/C_G(x)$) and the conjugacy class $c$ of $x$ 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.

Related facts

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. Group acts as automorphisms by conjugation
2. Fundamental theorem of group actions

Proof

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.