In group theory, a field of mathematics, a double coset is a collection of group elements which are equivalent under the symmetries coming from two subgroups, generalizing the notion of a single coset.
Suppose that G is a group with subgroups H and K acting by left and right multiplication, respectively.
The (H, K)-double cosets of G may be equivalently described as orbits for the product group H × K acting on G by (h, k) ⋅ x = hxk−1.
Many of the basic properties of double cosets follow immediately from the fact that they are orbits.
However, because G is a group and H and K are subgroups acting by multiplication, double cosets are more structured than orbits of arbitrary group actions, and they have additional properties that are false for more general actions.
Let H and K both act by right multiplication on G. Then G acts by left multiplication on the product of coset spaces G / H × G / K. The orbits of this action are in one-to-one correspondence with H \ G / K. This correspondence identifies (xH, yK) with the double coset Hx−1yK.
Briefly, this is because every G-orbit admits representatives of the form (H, xK), and the representative x is determined only up to left multiplication by an element of H. Similarly, G acts by right multiplication on H \ G × K \ G, and the orbits of this action are in one-to-one correspondence with the double cosets H \ G / K. Conceptually, this identifies the double coset space H \ G / K with the space of relative configurations of an H-coset and a K-coset.
Additionally, this construction generalizes to the case of any number of subgroups.
The analog of Lagrange's theorem for double cosets is false.
This means that the size of a double coset need not divide the order of G. For example, let G = S3 be the symmetric group on three letters, and let H and K be the cyclic subgroups generated by the transpositions (1 2) and (1 3), respectively.
If e denotes the identity permutation, then This has four elements, and four does not divide six, the order of S3.
As noted earlier, in this case the double coset space equals the left coset space G / HK.
Similarly, if K is normal, then H \ G / K is the right coset space HK \ G. Standard results about left and right coset spaces then imply the following facts.
Under certain finiteness conditions, there is a product on the free abelian group generated by the (H, K)- and (K, L)-double cosets with values in the free abelian group generated by the (H, L)-double cosets.
This means there is a bilinear function Assume for simplicity that G is finite.
This function may be pulled back along the projection G → H \ G / K which sends x to the double coset HxK.
By the way in which this function was constructed, it is left invariant under H and right invariant under K. The corresponding element of the group algebra Z[G] is and this element is invariant under left multiplication by H and right multiplication by K. Conceptually, this element is obtained by replacing HxK by the elements it contains, and the finiteness of G ensures that the sum is still finite.
It also follows that if M is a fourth subgroup of G, then the product of (H, K)-, (K, L)-, and (L, M)-double cosets is associative.
An important special case is when H = K = L. In this case, the product is a bilinear function This product turns Z[H \ G / H] into an associative ring whose identity element is the class of the trivial double coset [H].
Commutativity of the convolution product is closely tied to Gelfand pairs.
When the group G 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.
For instance, this happens for the Hecke algebra of a locally compact group.
reveals extra information about structure of the action of
acts transitively on the set of distinct pairs of
In the case of finite groups, this is Mackey's decomposition theorem.
They are also important in functional analysis, where in some important cases functions left-invariant and right-invariant by a subgroup K can form a commutative ring under convolution: see Gelfand pair.
In geometry, a Clifford–Klein form is a double coset space Γ\G/H, where G is a reductive Lie group, H is a closed subgroup, and Γ is a discrete subgroup (of G) that acts properly discontinuously on the homogeneous space G/H.
In number theory, the Hecke algebra corresponding to a congruence subgroup Γ of the modular group is spanned by elements of the double coset space
; the algebra structure is that acquired from the multiplication of double cosets described above.
(these have different properties depending on whether m and N are coprime or not), and the diamond operators