Difference between revisions of "Factor group"
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| − | <font color=" | + | <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.
- [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−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