Centralizer

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⁻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: C_4mm(4) = 4.
The set of symmetry operations of the point group 4mm 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 4mm is the subgroup mm2 obtained by taking the two mirror reflections normal to the tetragonal a and b axes: C_4mm(m_[100]) = mm2.