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}.
Example
- The set of symmetry operations of the point group 4mm which commute with 4^{1} is {1, 2, 4^{1} 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.