Difference between revisions of "Center"
From Online Dictionary of Crystallography
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| − | <font color="blue">Centre</font> (''Fr'') | + | <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''). |
| − | The ''' | + | 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 | + | The centre is an [[Abelian group]]. |
| − | The | + | 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'' → ''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'' → ''aga<sup>-1</sup>''. | ||
Revision as of 10:22, 13 May 2017
Centre (Fr). Zentrum (Ge). Centro (Sp). Centro (It). 中心 (Ja).
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: g → aga-1.