Normal subgroups are stable enough to be combined in several useful ways. Intersections of normal subgroups remain normal, and products of normal subgroups behave better than products of arbitrary subgroups. These results prepare the algebra needed for quotient groups and later for isomorphism theorems. In this lesson we work with $H\cap K$, $HK$, and $\langle H\cup K\rangle$, always checking whether the relevant set is actually a subgroup before discussing normality. A common mistake is to assume that $HK$ is automatically a subgroup for arbitrary subgroups $H$ and $K$; it is not automatic without an additional condition.

Let $(G,\circ)$ be a group and let $H,K$ be subgroups of $G$. The \textbf{product} of $H$ and $K$ is the subset $$\begin{equation*} HK=\{h\circ k:h\in H,\ k\in K\}. \end{equation*}$$