Product of subgroups

Symbol-free definition
The Product of two subgroups of a group is the subset consiting of the paarwise products between the two subgroups.

Definition with symbols
The Product $$HK$$ of two subgroups $$H$$ and $$K$$ of a group $$G$$ is:

$$HK := \{ hk \mid h \in H, k \in K \}$$.

If $$G$$ is abelian and if the group operation is denoted as $$+$$, the product is termed the sum, and is denoted $$H + K$$:

$$H + K = \{ h + k \mid h \in H, k \in K \}$$.

Facts
$$HK$$ is the double coset $$HeK$$, $$e$$ being the identity element of $$G$$.

The cardinality $$\left| HK \right|$$ of $$HK$$ is $$\left| H \right| \left| K \right| / \left| H \cap K \right|$$.

The product $$HK$$ is in general not a subgroup, because it may not be closed under the group operation.

The smallest subgroup containing $$HK$$ is the join $$\langle H, K \rangle$$ of $$H$$ and $$K$$, which is also the subgroup generated by $$H$$ and $$K$$.

Following statements are equivalent:


 * $$HK$$ is a subgroup
 * $$HK = \langle H,K \rangle$$, viz., it is precisely the join of $$H$$ and $$K$$ (the subgroup generated by $$H$$ and $$K$$)
 * $$\! HK = KH$$
 * $$HK \subseteq KH$$
 * $$KH \subseteq HK$$

If the above equivalent conditions hold, $$H$$ and $$K$$ are termed permuting subgroups.