double coset

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

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 $H\backslash G/K$ . It is straightforward to show that $H\backslash G/K$ is a 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, the $(H,K)$ -double cosets need not all be of the same cardinality. For example, if $G$ is the symmetric group $S_3$ , and $H=\genby{(1,2)}$ and $K=\genby{(1,3)}$ , then the two $(H,K)$ -double cosets are $\{e,(1,2),(1,3),(1,3,2)\}$ and $\{(2,3),(1,2,3)\}$ .