Product of two subgroups

Definition

Let $G$ be a group and let $H$ and $K$ be subgroups of $G$. The product of $H$ and $K$ is the subset $HK = \{hk : h \in H, k \in K\}$ of $G$.

Remarks

The set $HK$ is a subgroup of $G$ if and only if $HK = KH$. It need not be a subgroup. In the symmetric group $S_3$, two distinct subgroups of order $2$ have a product with $4$ elements, and $4$ does not divide $6$. When $HK$ is a subgroup, it is the subgroup generated by $H \cup K$.

Explanations

L1 human

To multiply two subgroups, multiply each element of the first by each element of the second. The result contains both subgroups, but it may fail to be a subgroup, because a product $hk h'k'$ need not have the form $h''k''$. In an abelian group, the product of two subgroups is always a subgroup.