1.37: Double coset
Let G be a group, and H and K be two subgroups of G. One says that the two elements g 1 ∈ G and g 2 ∈ G belong to the same double coset of G relative to H and K if there exist elements h i ∈ H and k j ∈ K such that:
g 2 = h i g 1 k j
The complex Hg 1 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 i ∈ G belongs to exactly one double coset. It is also a generalization of the coset decomposition, because the double coset Hg 1 K contains complete left cosets of K and complete right cosets of H.