Take a group $G$ with some partition $P$ such that for all $A,B\in P$, the product of the subsets $AB$ is contained entirely in some element $C$ of the partition. Let $N\in P$ contain 1. Prove $N$ is normal, and that $P$ is the set of cosets of $N$ in $G$. (Artin 2.10.3, trying to work through the whole book on my own)
It is pretty straightforward that $N$ is a normal subgroup. Take another $A \in P$. Because $N$ has the element $1$, then $A \subset AN$ and moreover $A = AN$ by the hypothesis.
We would be done with the proof if we could show $aN = AN$ for an element $a \in A$. Multiplying $A$ by any element in $N$ must preserve $A$, but why couldn't $A$ be a union of distinct cosets?