# Factor group

### From Online Dictionary of Crystallography

##### Revision as of 15:39, 10 April 2017 by MassimoNespolo (talk | contribs) (→See also: 6th edition of ITA)

Revision as of 15:39, 10 April 2017 by MassimoNespolo (talk | contribs) (→See also: 6th edition of ITA)

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

- Section 1.1.5 in the
*International Tables of Crystallography, Volume A*, 6^{th}edition