# 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 *i*.*e*. each g_{i} ∈ G belongs to exactly one double coset. It is also a generalization of the coset _{1}K contains