Double Coset

In mathematics, an (H,K) double coset in G, where G is a group and H and K are subgroups of G, is an equivalence class for the equivalence relation defined on G by

x ~ y if there are h in H and k in K with hxk = y.

Then each double coset is of the form HxK, and G is partitioned into its (H,K) double cosets; each of them is a union of ordinary right cosets Hy of H in G and left cosets zK of K in G. In another aspect, these are in fact orbits for the group action of H×K on G with H acting by left multiplication and K by inverse right multiplication. The set of double cosets can be written

H\G/K.