$HK$ is not a subgroup, such that $H,K$ do not commute.

Question

I know that if two subgroups commute, $HK=kH$, then $KH$ is a another subgroup. What's an example of $H,K$ do not commute such that $HK$ is not a subgroup.

Answer 1

Take the groups generated by non disjoint two cycles in $S_3$, say $H=\langle(12)\rangle$ and $K=\langle(23)\rangle$ in $S_3$. Then $$ |HK|=4\nmid |S_3|=6 $$ and thus is not a subgroup, by Lagrange's theorem.

Edit: as suggested, in case you do not know Lagrange's theorem, note that $$ HK=\{ (12),(23),(312)\} $$ which does not include $$ (312)^{-1}=(312)^2=(213) $$ (recall that the order of a cycle is it's length).