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.