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Difference between revisions of "Centralizer"

From Online Dictionary of Crystallography

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<font color="blue">Centralisateur</font> (''Fr''). <font color="red">Zentralisator</font> (''Ge''). <font color="black">Centralizzatore</font> (''It''). <font color="purple"> 中心化群 </font> (''Ja'').
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<font color="blue">Centralisateur</font> (''Fr''). <font color="red">Zentralisator</font> (''Ge''). <font color="black">Centralizzatore</font> (''It''). <font color="purple">中心化群</font> (''Ja''). <font color="green">Centralizador</font> (''Sp'').
  
  
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The '''centralizer''' ''C<sub>G</sub>(g)'' of an element ''g'' of a group ''G'' is the set of elements of ''G'' which commute with ''g'':
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: ''C<sub>G</sub>(g)'' = {''x'' &isin; ''G : xg = gx''}.
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If ''H'' is a [[subgroup]] of ''G'', then ''C<sub>H</sub>(g)'' = ''C<sub>G</sub>(g) ∩ H''.
  
The '''centralizer''' C<sub>G</sub>(g) of an element g of a group G is the set of elements of G which commute with g:
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More generally, if ''S'' is any subset of ''G'' (not necessarily a subgroup), the centralizer of ''S'' in ''G'' is defined as
: C<sub>G</sub>(g) = {x &isin; G : xg = gx}.
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: ''C<sub>G</sub>(S)'' = {''x'' &isin; ''G'' : &forall; ''s'' &isin; ''S'', ''xs = sx''}.
If H is a [[subgroup]] of G, then C<sub>H</sub>(g) = C<sub>G</sub>(g) ∩ H.
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If ''S'' = {''g''}, then ''C(S) = C(g)''.
  
More generally, if S is any subset of G (not necessarily a subgroup), the centralizer of S in G is defined as
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''C(S)'' is a subgroup of ''G''; in fact, if ''x, y'' are in ''C(S)'', then ''xy''<sup>&nbsp;&minus;1</sup>''s'' = ''xsy''<sup>&minus;1</sup> = ''sxy''<sup>&minus;1</sup>.
: C<sub>G</sub>(S) = {x &isin; G : &forall; s &isin; S, xs = sx}.
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If  S = {g}, then C(S) = C(g).
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==Example==
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* 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>&minus;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.
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* 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 ''mm''2 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.
  
C(S) is a subgroup of G; in fact, if x, y are in C(S), then ''xy''<sup>&nbsp;&minus;1</sup>''s'' = xsy<sup>&minus;1</sup> = sxy<sup>&minus;1</sup>.
 
  
 
==See also==
 
==See also==

Latest revision as of 17:01, 9 November 2017

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


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) = {xG : 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) = {xG : ∀ sS, 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