# Difference between revisions of "Centralizer"

### From Online Dictionary of Crystallography

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

+ | *[[Center]] | ||

*[[Normalizer]] | *[[Normalizer]] | ||

*[[Stabilizer]] | *[[Stabilizer]] | ||

[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] |

## Revision as of 10:47, 1 July 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}.