Difference between revisions of "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"> | + | <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''). |
==Definition== | ==Definition== | ||
| − | Let N be a [[normal subgroup]] of a group G. The '''factor group''' or '''quotient group''' | + | 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 [[coset]]s of N in G, i.e.: |
:<math>G/N = \{ aN : a \isin G \}.</math> | :<math>G/N = \{ aN : a \isin G \}.</math> | ||
Revision as of 13:55, 14 March 2015
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-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 8 in the International Tables of Crystallography, Volume A