# Factor group

(Difference between revisions)
 Revision as of 10:57, 15 May 2017 (view source)m (Style edits to align with printed edition)← Older edit Revision as of 15:16, 10 October 2017 (view source)m (two languages added)Newer edit → Line 1: Line 1: - Groupe facteur (''Fr''). Faktorgruppe (''Ge''). Grupo cociente (''Sp''). Gruppo fattore (''It''). 因子群 (商群、剰余群) (''Ja''). + 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== ==Definition==

## Revision as of 15:16, 10 October 2017

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.