Subgroup of a group

Question

Let H and K be subgroups of a group. Show that HK is a subgroup if and only if HK=KH.

Answer 1

First notice that if $HK$ is a subgroup of $G$ then it's closed under inverses so $HK = (HK)^{-1} = K^{-1}H^{-1} = KH$. Conversely if $HK = KH$ then take $hk$, $h^\prime k^\prime \in HK$. Then $(hk)(h^\prime k^\prime)^{-1} = hk(k^\prime)^{-1}(h^\prime)^{-1}$. Since $HK = KH$ we can rewrite $k(k^\prime)^{-1}(h^\prime)^{-1}$ as $h^{\prime \prime}k^{\prime \prime}$ for some new $h^{\prime \prime} \in H$, $k^{\prime \prime} \in K$. So $(hk)(h^\prime k^\prime)^{-1}=hh^{\prime \prime}k^{\prime \prime}$ which is in $HK$. This verifies that $HK$ is a subgroup.