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Difference between revisions of "Factor group"

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<font color="blue"> Groupe facteur</font> (''Fr''); <font color="black"> Gruppo fattore</font> (''It'').
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<font color="orange">زمرة خارج القسمة</font> (''Ar''). <font color="blue">Groupe facteur</font> (''Fr''). <font color="red">Faktorgruppe</font> (''Ge''). <font color="black">Gruppo fattore</font> (''It''). <font color="purple">因子群 (商群、剰余群)</font> (''Ja''). <font color="brown">Факторгруппа</font> (''Ru''). <font color="green">Grupo cociente</font> (''Sp'').  
  
 
==Definition==
 
==Definition==
Let N be a [[normal subgroup]] of a group G. The '''factor group''' or '''quotient group''' G/N is the set of all left [[coset]]s of N in G, i.e.:
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Let ''N'' be a [[normal subgroup]] of a group ''G''. The '''factor group''' or '''quotient group''' or '''residue class group''' ''G/N'' is the set of all left [[coset]]s of ''N'' in ''G'', ''i.e.''
  
 
:<math>G/N = \{ aN : a \isin G \}.</math>
 
:<math>G/N = \{ aN : a \isin G \}.</math>
  
For each aN and bN in G/N, the product of aN and bN is (aN)(bN), which is still a left coset. In fact, because N is normal:
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For each ''aN'' and ''bN'' in ''G/N'', the product of ''aN'' and ''bN'' is (''aN'')(''bN''), which is still a left coset. In fact, because ''N'' is normal:
  
:(aN)(bN) = a(Nb)N = a(bN)N = (ab)NN = (ab)N.
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:(''aN'')(''bN'') = ''a''(''Nb'')''N'' = ''a''(''bN'')''N'' = (''ab'')''NN'' = (''ab'')''N''.
  
The inverse of an element aN of G/N is a<sup>-1</sup>N.
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The inverse of an element ''aN'' of ''G/N'' is ''a''<sup>&minus;1</sup>''N''.
  
 
==Example==
 
==Example==
The factor group G/T of a space group G and its translation subgroup is the [[point group]] corresponding to G.
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The factor group ''G''/''T'' of a [[space group]] ''G'' and its translation subgroup is isomorphic to the [[point group]] ''P'' of ''G''.
  
 
==See also==
 
==See also==
Chapter 8 in the ''International Tables of Crystallography, Volume A''
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*Chapter 1.1.5 of ''International Tables for Crystallography, Volume A'', 6th edition
  
 
[[Category:Fundamental crystallography]]
 
[[Category:Fundamental crystallography]]

Latest revision as of 13:22, 13 November 2017

زمرة خارج القسمة (Ar). Groupe facteur (Fr). Faktorgruppe (Ge). Gruppo fattore (It). 因子群 (商群、剰余群) (Ja). Факторгруппа (Ru). Grupo cociente (Sp).

Definition

Let N be a normal subgroup of a group G. The factor group or quotient group or residue class group G/N is the set of all left cosets of N in G, i.e.

[math]G/N = \{ aN : a \isin G \}.[/math]

For each aN and bN in G/N, the product of aN and bN is (aN)(bN), which is still a left coset. In fact, because N is normal:

(aN)(bN) = a(Nb)N = a(bN)N = (ab)NN = (ab)N.

The inverse of an element aN of G/N is a−1N.

Example

The factor group G/T of a space group G and its translation subgroup is isomorphic to the point group P of G.

See also

  • Chapter 1.1.5 of International Tables for Crystallography, Volume A, 6th edition