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Difference between revisions of "Double coset"

From Online Dictionary of Crystallography

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<Font color="blue">Double coset</Font> (''Fr''). <Font color="black">Doppio coset</Font> (''It''). <Font color="purple">両側剰余類</Font> (''Ja'')
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Let G be a group, and H and K be two [[subgroup]]s of G. One says that the two elements g<sub>1</sub> &isin; G and g<sub>2</sub> &isin; G belong to the same '''double coset''' of G relative to H and K if there exist elements h<sub>i</sub> &isin; H and k<sub>j</sub> &isin; K such that:
 
Let G be a group, and H and K be two [[subgroup]]s of G. One says that the two elements g<sub>1</sub> &isin; G and g<sub>2</sub> &isin; G belong to the same '''double coset''' of G relative to H and K if there exist elements h<sub>i</sub> &isin; H and k<sub>j</sub> &isin; K such that:
  

Revision as of 13:36, 20 March 2015

Double coset (Fr). Doppio coset (It). 両側剰余類 (Ja)


Let G be a group, and H and K be two subgroups of G. One says that the two elements g1 ∈ G and g2 ∈ G belong to the same double coset of G relative to H and K if there exist elements hi ∈ H and kj ∈ K such that:

g2 = hig1kj

The complex Hg1K is called a double coset

The partition of G into double cosets relative to H and K is a classification, i.e. each gi ∈ G belongs to exactly one dobule coset. It is also a generalization of the coset decomposition, because the double coset Hg1K contains complete left cosets of K and complete right cosets of H.