Difference between revisions of "Factor group"
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| − | <font color="blue">Groupe facteur</font> (''Fr''). | + | <font color="blue">Groupe facteur</font> (''Fr''). <font color="red">Faktorgruppe</font> (''Ge''). <font color="green">Grupo cociente</font> (''Sp''). <font color="black">Gruppo fattore</font> (''It''). <font color="purple">因子群 (商群、剰余群)</font> (''Ja''). |
==Definition== | ==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. | + | 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: | + | 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. | + | :(''aN'')(''bN'') = ''a''(''Nb'')''N'' = ''a''(''bN'')''N'' = (''ab'')''NN'' = (''ab'')''N''. |
| − | The inverse of an element aN of G/N is a<sup> | + | The inverse of an element ''aN'' of ''G/N'' is ''a''<sup>−1</sup>''N''. |
==Example== | ==Example== | ||
| − | The factor group G/T of a [[space group]] G and its translation subgroup is isomorphic to the [[point group]] P of G. | + | 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 1.1.5 of ''International Tables for Crystallography, Volume A'', 6th edition |
[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] | ||
Revision as of 10:57, 15 May 2017
Groupe facteur (Fr). Faktorgruppe (Ge). Grupo cociente (Sp). Gruppo fattore (It). 因子群 (商群、剰余群) (Ja).
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