Double coset

From Online Dictionary of Crystallography

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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 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