Centralizer

Centralisateur (Fr); Zentralisator (Ge); Centralizzatore (It).

The centralizer C_G(g) of an element g of a group G is the set of elements of G which commute with g:

C_G(g) = {x ∈ G : xg = gx}.

If H is a subgroup of G, then C_H(g) = C_G(g) ∩ H.

More generally, if S is any subset of G (not necessarily a subgroup), the centralizer of S in G is defined as

C_G(S) = {x ∈ G : ∀ s ∈ S, xs = sx}.

If S = {g}, then C(S) = C(g).

C(S) is a subgroup of G; in fact, if x, y are in C(S), then xy ⁻¹s = xsy⁻¹ = sxy⁻¹.

See also

Normalizer