# Factor group

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.

$G/N = \{ aN : a \isin G \}.$

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.