Double coset

In group theory, a field of mathematics, a double coset is a collection of group elements which are equivalent under the symmetries coming from two subgroups. More precisely, let G be a group, and let H and K be subgroups. Let H act on G by left multiplication and let K act on G by right multiplication. For each x in G, the (H, K)-double coset of x is the set

${\displaystyle HxK=\{hxk\colon h\in H,k\in K\}.}$

When H = K, this is called the H-double coset of x. Equivalently, HxK is the equivalence class of x under the equivalence relation

x ~ y if and only if there exist h in H and k in K such that hxk = y.

The set of all ${\displaystyle (H,K)}$-double cosets is denoted by ${\displaystyle H\,\backslash G/K.}$