Double coset

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 h_i ∈ H and k_j ∈ K such that:

g₂ = h_ig₁k_j

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 g_i ∈ 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.