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

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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

Other related facts/indirect applications

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.