Having learned how cosets partition a group and how indices measure the size of that partition, the next structural question is whether cosets can themselves be multiplied without ambiguity. This question leads naturally to normal subgroups. A subgroup may sit inside a group in many ways, but a normal subgroup sits symmetrically: its left cosets and right cosets coincide. Normal subgroups are the subgroups from which quotient groups are built, so they form one of the central bridges from elementary group theory to homomorphisms and isomorphism theorems. Students often think that every subgroup behaves well with cosets; the first purpose of this lesson is to separate ordinary subgroup behaviour from normal subgroup behaviour.

Let $(G,\circ)$ be a group and let $H$ be a subgroup of $G$. Then $H$ is called a \textbf{normal subgroup} of $G$ if $$\begin{equation*} xH=Hx \quad \forall x\in G. \end{equation*}$$ It is denoted by $$\begin{equation*} H\trianglelefteq G. \end{equation*}$$