From Online Dictionary of Crystallography
Revision as of 13:54, 10 November 2017 by BrianMcMahon (Tidied translations and added German and Spanish (U. Mueller))
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 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.