Difference between revisions of "Double coset"
From Online Dictionary of Crystallography
BrianMcMahon (talk | contribs) (Tidied translations and added German and Spanish (U. Mueller)) |
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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.e.'' each ''g<sub>i</sub>'' ∈ ''G'' belongs to exactly one | + | 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 === | === See also === | ||
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.