Factor group

From Online Dictionary of Crystallography

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<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'').
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<font color="orange">زمرة خارج القسمة</font> (''Ar''); <font color="blue">Groupe facteur</font> (''Fr''); <font color="red">Faktorgruppe</font> (''Ge''); <font color="black">Gruppo fattore</font> (''It''); <font color="purple">因子群 (商群、剰余群)</font> (''Ja''); <font color="brown">Факторгруппа</font> (''Ru''); <font color="green">Grupo cociente</font> (''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.

See also

  • Chapter 1.1.5 of International Tables for Crystallography, Volume A, 6th edition