# Product of subgroups

## Definition

### 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.