Left coset space of centralizer is in bijective correspondence with conjugacy class
From Groupprops
Contents
Statement
For a group and an element
in
, there is a bijection between the space of left cosets of the centralizer
in
(denoted
) and the conjugacy class
of
in
.
In particular:
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
Facts used
Proof
Proof outline
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.