Product of subgroups

Definition

Symbol-free definition

The Product of two subgroups of a group is the subset consiting of the pairwise 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.