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Groupprops β

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

Facts used

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.