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. More precisely, let be a group, and let and be subgroups. Let act on by left multiplication and let act on by right multiplication. For each in , the -double coset of is the set When , this is called the -double coset of . Equivalently, is the equivalence class of under the equivalence relation if and only if there exist in and in such that . The set of all double cosets is denoted by Properties Suppose that is a group with subgroups and acting by left and right multiplication, respectively. The -double cosets of may be equivalently described as orbits for the product group acting on by . Many of the basic properties of double cosets follow immediately from the fact that they are orbits. However, because is a group and and 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. Two double cosets and are either disjoint or identical. is the disjoint union of its double cosets. There is a one-to-one correspondence between the two double coset spaces and given by identifying with . If , then . If , then . A double coset is a union of right cosets of and left cosets of ; specifically, The set of -double cosets is in bijection with the orbits , and also with the orbits under the mappings and respectively. If is normal, then is a group, and the right action of on this group factors through the right action of . It follows that . Similarly, if is normal, then . If is a normal subgroup of , then the -double cosets are in one-to-one correspondence with the left (and right) -cosets. Consider as the union of a -orbit of right -cosets. The stabilizer of the right -coset with respect to the right action of is . Similarly, the stabilizer of the left -coset with respect to the left action of is . It follows that the number of right cosets of contained in is the index and the number of left cosets of contained in is the index . Therefore If , , and are finite, then it also follows