Note that the -double cosets are just the right cosets of , and the -double cosets are just the left cosets of . In general, every -double coset is a union of right cosets of , and also a union of left cosets of .
The set of all -double cosets is denoted . It is straightforward to show that is a partition (http://planetmath.org/Partition) of , that is, every element of lies in exactly one -double coset.
In contrast to the situation with ordinary cosets (http://planetmath.org/Coset), the -double cosets need not all be of the same cardinality. For example, if is the symmetric group (http://planetmath.org/SymmetricGroup) , and and , then the two -double cosets are and .