  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 \mid $a
\in A$ and $b\in B$}$ satisfies $|A+B| < |A| + |B|$. Vosper proved a theorem
which characterizes all critical pairs in the special case when $|G|$ 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.
