. Standard results about left and right coset spaces then imply the following facts. for all in . That is, all double cosets have the same cardinality. If is finite, then . In particular, and divide . Examples Let be the symmetric group, considered as permutations of the set }. Consider the subgroup which stabilizes . Then consists of two double cosets. One of these is , and the other is for any permutation which does not fix . This is contrasted with , which has elements , where each . Let be the group , and let be the subgroup of upper triangular matrices. The double coset space is the Bruhat decomposition of . The double cosets are exactly , where ranges over all n-by-n permutation matrices. For instance, if , then Products in the free abelian group on the set of double cosets Suppose that is a group and that , , and are subgroups. Under certain finiteness conditions, there is a product on the free abelian group generated by the - and -double cosets with values in the free abelian group generated by the -double cosets. This means there is a bilinear function Assume for simplicity that is finite. To define the product, reinterpret these free abelian groups in terms of the group algebra of as follows. Every element of has the form where is a set of integers indexed by the elements of . This element may be interpreted as a -valued function on , specifically, . This function may be pulled back along the projection which sends to the double coset . This results in a function . By the way in which this function was constructed, it is left invariant under and right invariant under . The corresponding element of the group algebra is and this element is invariant under left multiplication by and right multiplication by . Conceptually, this element is obtained by replacing by the elements it contains, and the finiteness of ensures that the sum is still finite. Conversely, every element of which is left invariant under and right invariant under is the pullback of a function on . Parallel statements are true for and . When elements of , , and are interpreted as invariant elements of , then the product whose existence was asserted above is precisely the multiplication in . Indeed, it is trivial to check that the