# Difference between revisions of "Factor group"

### From Online Dictionary of Crystallography

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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*^{−1}*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