# Double coset of a pair of subgroups

(Redirected from Double coset space)

## Definition

### Definition with symbols

Let $H$ and $K$ be subgroups of a group $G$. Then a subset $L$ of $G$ is termed a double coset for $H$ and $K$ if the following equivalent conditions are satisfied:

• There exists an element $x$ in $G$ such that $L = HxK$
• For any element $x$ in $L$, $L = HxK$.

## 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 $M$ be a subgroup of $G$. We can consider the following three special cases:

• $H = M$ and $K$ is trivial. In this case, the double cosets of $H$ and $K$ are the same as the right cosets of $M$
• $H$ is trivial and $K = M$. In this case, the double cosets of $H$ and $K$ are the same as the left cosets of $M$
• $H = K = M$. In this case, the double cosets of $H$ and $K$ are simply called the double cosets of $M$.

### 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 if and only if the subgroup is normal.

### Double coset cardinality

The cardinality $\left| HxK \right|$ of the double coset $HxK$ is $\left| H \right| \left| K \right| / \left| H \cap xKx^{-1} \right|$. $HxK$ is the union of $[H:H \cap xKx^{-1}] = \left| HxK \right| / \left| K \right|$ different (and hence disjoint) left cosets of $K$. $[G:K] = \sum_{x \in X} [H:H \cap xKx^{-1}]$ where $X$ is a set of representatives of the double cosets of $H$ and $K$.

### 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 $HxK \le HyK$ if there exist $g,h \in G$ such that $gHxKh \subseteq HyK$.

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 spaces $G/H$ and $G/K$. $G$ acts on both these coset spaces by left multiplication, and we can hence consider the action of $G$ on the product of the coset spaces $G/H \times G/K$. The orbits under this action are the double cosets of $H$ and $K$ in $G$.

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

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.