Product of subgroups

Definition

Let $G$ be a group and $H,K$ be subgroups of $G$. Then the product of $H$ and $K$ is defined as the set of elements:

$\{hk|h\in H,k\in K\}$

The following are equivalent (but may not all in general be true):

• $HK$ 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.