# Double coset of a pair of subgroups

## Contents

## Definition

### Definition with symbols

Let and be subgroups of a group . Then a subset of is termed a **double coset** for and if the following equivalent conditions are satisfied:

- There exists an element in such that
- For any element in , .

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

- and is trivial. In this case, the double cosets of and are the same as the right cosets of

- is trivial and . In this case, the double cosets of and are the same as the left cosets of

- . In this case, the double cosets of and are simply called the double cosets of .

### 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.

### 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 if there exist such that .

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

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

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.