# Difference between revisions of "Centralizer"

### From Online Dictionary of Crystallography

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C(S) is a subgroup of G; in fact, if x, y are in C(S), then ''xy''<sup> −1</sup>''s'' = xsy<sup>−1</sup> = sxy<sup>−1</sup>. | C(S) is a subgroup of G; in fact, if x, y are in C(S), then ''xy''<sup> −1</sup>''s'' = xsy<sup>−1</sup> = sxy<sup>−1</sup>. | ||

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+ | ==Example== | ||

+ | * The set of symmetry operations of the point group 4''mm'' which commute with 4<sup>1</sup> is {1, 2, 4<sup>1</sup> and 4<sup>-1</sup>}. The centralizer of the fourfold positive rotation with respect to the point group 4''mm'' is the subgroup 4: C<sub>4''mm''</sub>(4) = 4. | ||

+ | * The set of symmetry operations of the point group 4''mm'' which commute with m<sub>[100]</sub> is {1, 2, m<sub>[100]</sub> and m<sub>[010]</sub>}. The centralizer of the m<sub>[100]</sub> reflection with respect to the point group 4''mm'' is the subgroup mm2 obtained by taking the two mirror reflections normal to the tetragonal '''a''' and '''b''' axes: C<sub>4''mm''</sub>(m<sub>[100]</sub>) = ''mm''2. | ||

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==See also== | ==See also== |

## Revision as of 13:55, 27 August 2014

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 4
*mm*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 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 reflections normal to the tetragonal**a**and**b**axes: C_{4mm}(m_{[100]}) =*mm*2.