View source for Factor group

<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==
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>

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''<sup>&minus;1</sup>''N''.

==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

[[Category:Fundamental crystallography]]