product of a left--invariant element and a right--invariant element continues to be left--invariant and right--invariant. The bilinearity of the product follows immediately from the bilinearity of multiplication in . It also follows that if is a fourth subgroup of , then the product of -, -, and -double cosets is associative. Because the product in corresponds to convolution of functions on , this product is sometimes called the convolution product. An important special case is when . In this case, the product is a bilinear function This product turns into an associative ring whose identity element is the class of the trivial double coset . In general, this ring is non-commutative. For example, if , then the ring is the group algebra , and a group algebra is a commutative ring if and only if the underlying group is abelian. If is normal, so that the -double cosets are the same as the elements of the quotient group , then the product on is the product in the group algebra . In particular, it is the usual convolution of functions on . In this case, the ring is commutative if and only if is abelian, or equivalently, if and only if contains the commutator subgroup of . If is not normal, then may be commutative even if is non-abelian. A classical example is the product of two Hecke operators. This is the product in the Hecke algebra, which is commutative even though the group is the modular group, which is non-abelian, and the subgroup is an arithmetic subgroup and in particular does not contain the commutator subgroup. Commutativity of the convolution product is closely tied to Gelfand pairs. When the group is a topological group, it is possible to weaken the assumption that the number of left and right cosets in each double coset is finite. The group algebra is replaced by an algebra of functions such as or , and the sums are replaced by integrals. The product still corresponds to convolution. For instance, this happens for the Hecke algebra of a locally compact group. Applications When a group has a transitive group action on a set , computing certain double coset decompositions of reveals extra information about structure of the action of on . Specifically, if is the stabilizer subgroup of some element , then decomposes as exactly two double cosets of if and only if acts transitively