Centralizer and normalizer

In group theory, which is a branch of abstract algebra, the concepts of centralizer and normalizer help us understand the structure of groups and their subgroups. Here are the definitions of both: ### Centralizer The centralizer of a subset $ S $ of a group $ G $, denoted as $ C_G(S) $, is the set of all elements in $ G $ that commute with every element of $ S $.

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Centralizer and normalizer by

Codex  0 Created 2026-09-24 Updated 2026-10-14

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Both constructions are stabilizer subgroups for the conjugation action. The centralizer

C_{G} (x)

fixes an individual element

x

; the normalizer

N_{G} (H)

preserves a subgroup

H

as a whole, while possibly moving its elements. The centralizer of a subset

C_{G} (H)

fixes every member of

H

individually, so

C_{G} (H) \subseteq N_{G} (H) .

(1)

For a cyclic subgroup

H = ⟨ r ⟩

, commuting with the group generator

r

is equivalent to commuting with all its powers, giving

C_{G} (H) = C_{G} (r)

In the symmetric group

S_{3}

, let

r = (123)

and

H = ⟨ r ⟩ = {e, (123), (132)}

. Then

C_{S_{3}} (r) = C_{S_{3}} (H) = H, N_{S_{3}} (H) = S_{3} .

(2)

Every transposition sends

r

r^{- 1}

under conjugation. In particular, for

t = (12)

t r t^{- 1} = r^{- 1} \neq = r, t H t^{- 1} = {e, r^{- 1}, r} = H .

(3)

Thus

t

belongs to the normalizer of

H

but not to the centralizer of

r

. Preserving the collection of powers allows the two possible directions of its group generator to be exchanged.

Figure 1.
The centralizer fixes $r = (123)$ , while the normalizer preserves $H = ⟨ r ⟩$ and may exchange $r$ with its inverse element
. **Details**.

This distinction lets one ask separately which operations leave a chosen operation unchanged and which operations preserve the subgroup it generates. The normalizer also supplies a group homomorphism to the automorphism group of