View source for Double coset of a pair of subgroups

==Definition==

===Definition with symbols===

Let <math>H</math> and <math>K</math> be [[subgroup]]s of a [[group]] <math>G</math>. Then a subset <math>L</math> of <math>G</math> is termed a '''double coset''' for <math>H</math> and <math>K</math> if the following equivalent conditions are satisfied:

* There exists an element <math>x</math> in <math>G</math> such that <math>L = HxK</math>
* For any element <math>x</math> in <math>L</math>, <math>L = HxK</math>.

==Facts==

===Equivalence relation===

The double cosets of a pair of subgroups are pairwise disjoint and hence form a partition of the group. The relation of being in the ''same'' double coset is an equivalence relation on the elements of the group.

===Special cases===

Let <math>M</math> be a subgroup of <math>G</math>. We can consider the following three special cases:

* <math>H = M</math> and <math>K</math> is [[trivial group|trivial]]. In this case, the double cosets of <math>H</math> and <math>K</math> are the same as the [[right coset]]s of <math>M</math>

* <math>H</math> is [[trivial group|trivial]] and <math>K = M</math>. In this case, the double cosets of <math>H</math> and <math>K</math> are the same as the [[left coset]]s of <math>M</math>

* <math>H = K = M</math>. In this case, the double cosets of <math>H</math> and <math>K</math> are simply called the double cosets of <math>M</math>.

===For a normal subgroup===

For a [[normal subgroup]], the notions of left coset, right coset, and double coset are equivalent.

===Double coset index===

The double coset index of a pair of subgroups is the number of double cosets.

The double coset index of a subgroup is the number of double cosets it has as a subgroup (that is, where both subgroups are equal to the given subgroup).

Note that the double coset index equals the usual [[index of a subgroup|index]] if and only if the subgroup is normal.

[[Bound on double coset index in terms of orders of group and subgroup]].

===Double coset cardinality===

The cardinality <math>\left| HxK \right|</math> of the double coset <math>HxK</math> is
<math>\left| H \right| \left| K \right| / \left| H \cap xKx^{-1} \right| </math>.

<math>HxK</math> is the union of
<math>
    [H:H \cap xKx^{-1}]
  = \left| HxK \right| / \left| K \right|
</math> different (and hence disjoint) left cosets of <math>K</math>.

<math>
    [G:K] = \sum_{x \in X} [H:H \cap xKx^{-1}]
</math>
where <math>X</math> is a set of representatives of the double cosets of <math>H</math> and <math>K</math>.

===Quasiorder on a collection of double cosets===

We can order the double cosets of a pair of subgroups by the [[translate-containment quasiorder]], wherein we say <math>HxK \le HyK</math> if there exist <math>g,h \in G</math> such that <math>gHxKh \subseteq HyK</math>.

Under this quasiorder, all the double cosets that actually comprise single cosets are the ''smallest'' or minimal elements -- these are in fact precisely the single cosets inside the [[normalizer]] of the subgroup. A pair of subgroups for which this quasiorder is actually a partial order (that is, two distinct double cosets cannot both be translate-contained in the other) is termed a [[double coset-ordering subgroup pair]]. When both member subgroups are the same, we call it a [[double coset-ordering subgroup]].

==Double coset space in terms of orbits under a group action==

Consider the product of the left [[coset space]]s <math>G/H</math> and <math>G/K</math>. <math>G</math> acts on both these coset spaces by left multiplication, and we can hence consider the action of <math>G</math> on the product of the coset spaces <math>G/H \times G/K</math>. The orbits under this action are the double cosets of <math>H</math> and <math>K</math> in <math>G</math>.

The proof of this comes from the fact that for every orbit, we can choose a representative where the first coordinate is the coset <math>H</math> itself. In this case, the second representative gives a coset of <math>K</math>. However, this coset is ambiguous upto left multiplication by an element of <math>H</math>. So the upshot is that we get a union of left cosets of <math>K</math>, under the left action of <math>H</math>. This is a double coset of <math>H</math> and <math>K</math>.

Double cosets thus measure the ''relative position'' of two left cosets.

This alternative approach to double cosets also allows us to generalize to the notion of a [[multicoset for a tuple of subgroups]].