# Centralizer

### From Online Dictionary of Crystallography

Centralisateur (*Fr*). Zentralisator (*Ge*). Centralizzatore (*It*). 中心化群 (*Ja*).

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*^{ −1}*s* = xsy^{−1} = sxy^{−1}.