# Centralizer

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

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

CG(g) = {xG : xg = gx}.

If H is a subgroup of G, then CH(g) = CG(g) ∩ H.

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

CG(S) = {xG : ∀ sS, 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.

## Example

• The set of symmetry operations of the point group 4mm which commute with 41 is {1, 2, 41 and 4−1}. The centralizer of the fourfold positive rotation with respect to the point group 4mm is the subgroup 4: C4mm(4) = 4.
• The set of symmetry operations of the point group 4mm which commute with m is {1, 2, m and m}. The centralizer of the m reflection with respect to the point group 4mm is the subgroup mm2 obtained by taking the two mirror reflections normal to the tetragonal a and b axes: C4mm(m) = mm2.