A New Proof of Kemperman's Theorem

Let $G$ be an additive abelian group and let $A,B\subseteq G$ be finite and nonempty. The pair $(A,B)$ is called critical if the sumset A+B = {a+b \mida \in A$and$b\in B} satisfies $\mathrm{\mid }A+B\mathrm{\mid }<\mathrm{\mid }A\mathrm{\mid }+\mathrm{\mid }B\mathrm{\mid }$. Vosper proved a theorem which characterizes all critical pairs in the special case when $\mathrm{\mid }G\mathrm{\mid }$ is prime. Kemperman generalized this by proving a structure theorem for critical pairs in an arbitrary abelian group. Here we give a new proof of Kemperman’s Theorem.