Difference between revisions of "Centralizer"
From Online Dictionary of Crystallography
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[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] | ||
Revision as of 15:45, 27 February 2007
Centralisateur (Fr); Zentralisator (Ge); Centralizzatore (It).
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.