For subgroups $H$ and $K$ of a group $G$, under what conditions is $HK$ a normal subgroup of $G$?
I know that $H\le N_G(K)$ is a sufficient but not necessary condition for $HK$ to be a subgroup of $G$, and that if both $H$ and $K$ are normal in $G$, then $HK$ is a normal subgroup of $G$ as well. But I'm looking for a weaker condition for $HK$ to be normal.
First of all, if $H$ and $K$ are subgroups, $HK$ need not be a subgroup. In fact, it is a subgroup if and only if $HK=KH$, see also here: $HK$ is a subgroup of $G$ if and only if $HK = KH$. If one of the subgroups is normal, this is automatically satisfied and $HK$ is a subgroup - see Proving that HK is a subgroup when K is normal.
Secondly, it looks like that the only concise condition for $HK$ being a normal subgroup is, that both $H$ and $K$ are normal subgroups.