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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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<Font color="blue">Double coset</Font> (''Fr''). <Font color="black">Doppio coset</Font> (''It''). <Font color="purple">両側剰余類</Font> (''Ja'').
  
  
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:
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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
  
 
<div align="center">
 
<div align="center">
g<sub>2</sub> = h<sub>i</sub>g<sub>1</sub>k<sub>j</sub>
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''g''<sub>2</sub> = ''h<sub>i</sub>g''<sub>1</sub>''k<sub>j</sub>''.
 
</div>
 
</div>
  
The [[complex]] Hg<sub>1</sub>K is  called a '''double coset'''
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The [[complex]] ''Hg''<sub>1</sub>''K'' is  called a '''double coset'''
  
The partition of G into double cosets relative to H and K is a classification, ''i''.''e''. each g<sub>i</sub> &isin; G belongs to exactly one dobule coset. It is also a generalization of the [[coset]] decomposition, because the double coset Hg<sub>1</sub>K contains complete left cosets of K and complete right cosets of H.
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The partition of ''G'' into double cosets relative to ''H'' and ''K'' is a classification, ''i.e.'' each ''g<sub>i</sub>'' &isin; ''G'' belongs to exactly one dobule coset. It is also a generalization of the [[coset]] decomposition, because the double coset ''Hg''<sub>1</sub>''K'' contains complete left cosets of ''K'' and complete right cosets of ''H''.
  
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=== See also ===
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*[[Coset]]
  
 
[[Category:Fundamental crystallography]]
 
[[Category:Fundamental crystallography]]

Revision as of 13:26, 13 May 2017

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 g1G and g2G belong to the same double coset of G relative to H and K if there exist elements hiH and kjK 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 giG 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.

See also