double coset

Let $H$ and $K$ be subgroups of a group $G$.An $\mathrm{(}H\mathrm{,}K\mathrm{)}$-double coset is a set of the form $HxK$ for some $x\in G$.Here $HxK$ is defined in the obvious way as

$$HxK=\{hxk\mid h\in H\text{ and }k\in K\}.$$

Note that the $(H,\{1\})$-double cosets are just the right cosets of $H$,and the $(\{1\},K)$-double cosets are just the left cosets of $K$.In general, every $(H,K)$-double coset is a union of right cosets of $H$,and also a union of left cosets of $K$.

The set of all $(H,K)$-double cosets is denoted $H\backslash \backslash G/K$.It is straightforward to show that $H\backslash \backslash G/K$ is a partition (http://planetmath.org/Partition) of $G$,that is, every element of $G$ lies in exactly one $(H,K)$-double coset.

In contrast to the situation with ordinary cosets (http://planetmath.org/Coset),the $(H,K)$-double cosets need not all be of the same cardinality.For example, if $G$ is the symmetric group (http://planetmath.org/SymmetricGroup) $S_3$,and $H=⟨(1,2)⟩$ and $K=⟨(1,3)⟩$,then the two $(H,K)$-double cosetsare $\{e,(1,2),(1,3),(1,3,2)\}$ and $\{(2,3),(1,2,3)\}$.