Centralizer

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


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) = {x ∈ G : 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) = {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: C4mm(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: C4mm(m[100]) = mm2.


See also