Double coset of a pair of subgroups

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 $L$, $L=HxL$.

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 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.