# Factor group

### From Online Dictionary of Crystallography

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*^{−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