Normality is always relative to an ambient group. A subgroup may be normal in one group but not in a larger group, and this is why the notation $H\trianglelefteq G$ must mention both $H$ and $G$. In this lesson we examine what happens when the surrounding group is replaced by an intermediate subgroup, by a product subgroup, or by a subgroup that intersects a normal subgroup. These results are often used silently later in quotient group arguments, so it is useful to prove them explicitly now.

Let $(G,\circ)$ be a group and let $H,L$ be subgroups of $G$. The subgroup $L$ is called an \textbf{intermediate subgroup between $H$ and $G$} if $$\begin{equation*} H\subseteq L\subseteq G. \end{equation*}$$