The product $HK$ is a subgroup when $H$ normalizes $K$

Let \(G\) be a group and \(H, K \leq G\) subgroups with \(H \leq N_G(K)\), where \(N_G(K)\) is the normalizer of \(K\). Then \(HK = \theset{hk \st h \in H,\ k \in K}\) is a subgroup of \(G\). In particular, this holds when \(K \normal G\).