Difference between revisions of "Double coset"
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| − | < | + | <font color="blue">Double coset</font> (''Fr''). <font color="red">Doppelte Nebenklasse</font> (''Ge''). <font color="black">Doppio coset</font> (''It''). <font color="purple">両側剰余類</font> (''Ja''). <font color="green">Clase lateral doble</font> (''Sp''). |
| − | 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> ∈ G and g<sub>2</sub> ∈ G belong to the same '''double coset''' of G relative to H and K if there exist elements h<sub>i</sub> ∈ H and k<sub>j</sub> ∈ 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> ∈ ''G'' and ''g''<sub>2</sub> ∈ ''G'' belong to the same '''double coset''' of ''G'' relative to ''H'' and ''K'' if there exist elements ''h<sub>i</sub>'' ∈ ''H'' and ''k<sub>j</sub>'' ∈ ''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> | + | ''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''' | + | 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 | + | The partition of ''G'' into double cosets relative to ''H'' and ''K'' is a classification, ''i.e.'' each ''g<sub>i</sub>'' ∈ ''G'' belongs to exactly one double 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''. |
| + | === See also === | ||
| + | |||
| + | *[[Coset]] | ||
[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] | ||
Latest revision as of 15:36, 23 July 2024
Double coset (Fr). Doppelte Nebenklasse (Ge). Doppio coset (It). 両側剰余類 (Ja). Clase lateral doble (Sp).
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 double 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.