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Difference between revisions of "Center"

From Online Dictionary of Crystallography

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<font color="blue">Centre</font> (''Fr''); <font color="red">Zentrum</font> (''Ge''); <font color="green">Centro</font> (''Sp''); <font color="black">Centro</font> (''It''); <font color="purple">中心</font> (''Ja'').
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<font color="orange">مركز</font> (''Ar''). <font color="blue">Centre</font> (''Fr''). <font color="red">Zentrum</font> (''Ge''). <font color="black">Centro</font> (''It''). <font color="purple">中心</font> (''Ja''). <font color="green">Centro</font> (''Sp'').
  
  
The '''center''' (or '''centre''') of a [[group]] ''G'' is the set ''Z(G)'' =  
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The '''centre''' (or '''center''') of a [[group]] ''G'' is the set ''Z(G)'' =  
 
{ ''a'' in ''G'' : ''a*g'' = ''g*a'' for all ''g'' in ''G'' } of elements commuting with all elements of ''G''.  
 
{ ''a'' in ''G'' : ''a*g'' = ''g*a'' for all ''g'' in ''G'' } of elements commuting with all elements of ''G''.  
The center is an [[Abelian group]].
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The centre is an [[Abelian group]].
  
The center of a group ''G'' is always a [[normal subgroup]] of ''G'', namely the [[group homomorphism|kernel]]  
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The centre of a group ''G'' is always a [[normal subgroup]] of ''G'', namely the [[group homomorphism|kernel]]  
 
of the [[group homomorphism|homomorphism]] mapping an element ''a'' of ''G'' to the [[automorphism|inner automorphism]] ''f<sub>a</sub>'': ''g'' &rarr; ''aga<sup>-1</sup>''.  
 
of the [[group homomorphism|homomorphism]] mapping an element ''a'' of ''G'' to the [[automorphism|inner automorphism]] ''f<sub>a</sub>'': ''g'' &rarr; ''aga<sup>-1</sup>''.  
  
 
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==See also==
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*[[Centralizer]]
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*[[Normalizer]]
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*[[Stabilizer]]
  
  
 
[[Category:Fundamental crystallography]]
 
[[Category:Fundamental crystallography]]

Latest revision as of 17:00, 9 November 2017

مركز (Ar). Centre (Fr). Zentrum (Ge). Centro (It). 中心 (Ja). Centro (Sp).


The centre (or center) of a group G is the set Z(G) = { a in G : a*g = g*a for all g in G } of elements commuting with all elements of G. The centre is an Abelian group.

The centre of a group G is always a normal subgroup of G, namely the kernel of the homomorphism mapping an element a of G to the inner automorphism fa: gaga-1.

See also