Let \(G\) be a group, and let \(H, K \le G\) be subgroups of \(G\). Prove that the set product \(H K = \{h k \mid h \in H, \; k \in K\}\) is a subgroup of \(G\) if and only if \(H K = K H\).
Solution
For subgroups \(H,K\le G\), \[ (HK)^{-1}=K^{-1}H^{-1}=KH. \] Hence if \(HK\) is a subgroup, then it is closed under inverses and therefore \[ HK=(HK)^{-1}=KH. \]
Conversely, suppose \(HK=KH\). Then \(e\in HK\), and \[ (HK)(HK)=H(KH)K=H(HK)K=(HH)(KK)=HK, \] so \(HK\) is closed under multiplication. Also \[ (HK)^{-1}=KH=HK, \] so it is closed under inverses. Therefore \(HK\le G\).