# Centralizer

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

- 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}.

### Example

- The set of symmetry operations of the point group 4
*mm*which commute with 4^{1}is {1, 2, 4^{1}and 4^{-1}}. The centralizer of the fourfold positive rotation withrespect to the point group 4*mm*is the subgroup 4: C_{4mm}(4) = 4. - The set of symmetry operations of the point group 4
*mm*which commute with m_{[100]}is {1, 2, m_{[100]}and m_{[010]}}. The centralizer of the m_{[100]}reflection with respect to the point group 4*mm*is the subgroup mm2 obtained by taking the two mirror reflectionsnormal to the tetragonal**a**and**b**axes: C_{4mm}(m_{[100]}) =*mm*2.