Centralizer - Revision history

BrianMcMahon: Tidied translations and added Spanish (U. Mueller)

2017-11-09T17:01:09Z

Tidied translations and added Spanish (U. Mueller)

BrianMcMahon: Style edits to align with printed edition

2017-05-13T10:27:57Z

Style edits to align with printed edition

MassimoNespolo: example

2014-08-27T13:55:24Z

example

MassimoNespolo: link

2014-07-01T10:47:34Z

link

MassimoNespolo at 14:46, 24 April 2007

2007-04-24T14:46:53Z

MassimoNespolo: link

2007-02-27T15:45:02Z

link

MassimoNespolo at 14:53, 27 February 2007

2007-02-27T14:53:07Z

New page

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'' −1''s'' = xsy−1 = sxy−1.

==See also==
[[Normalizer]]

[[Category:Fundamental crystallography]]

← Older revision		Revision as of 17:01, 9 November 2017
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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="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'').

@@ Line 2: / Line 2: @@
-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:
+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}.
+: ''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.
+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
+''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>.
 ==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 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.
-* 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.
+* 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.

← Older revision		Revision as of 13:55, 27 August 2014
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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>.
		+
		+	==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.
		+

	==See also==		==See also==

@@ Line 14: / Line 14: @@
 ==See also==
 *[[Normalizer]]
 *[[Stabilizer]]
 [[Category:Fundamental crystallography]]

← Older revision		Revision as of 14:46, 24 April 2007
Line 1:		Line 1:
−	<font color="blue">Centralisateur</font> (''Fr''); <font color="red">Zentralisator</font> (''Ge''); <font color="black">Centralizzatore</font> (''It'').	+	<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'').