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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> (''Ge''); <font color="black">Gruppo fattore</font> (''It'').
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<font color="blue">Groupe facteur</font> (''Fr'').; <font color="red">Faktorgruppe</font> (''Ge''). <font color="green">Grupo cociente</font> (''Ge''). <font color="black">Gruppo fattore</font> (''It''). <font color="purple">剰余群(商群)</font> (''Ja'').
  
 
==Definition==
 
==Definition==

Revision as of 10:16, 8 May 2007

Groupe facteur (Fr).; Faktorgruppe (Ge). Grupo cociente (Ge). Gruppo fattore (It). 剰余群(商群) (Ja).

Definition

Let N be a normal subgroup of a group G. The factor group or quotient 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 the point group corresponding to G.

See also

Chapter 8 in the International Tables of Crystallography, Volume A