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

From Groupprops

(Redirected from Size of conjugacy class equals index of centralizer)

## 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.