A short proof of Kneser's addition theorem for abelian groups

Abstract

Martin Kneser proved the following addition theorem for every abelian group $G$. If $A,B\subseteq G$ are finite and nonempty, then $|A+B|\ge |A+K|+|B+K|-|K|$ where $K=\{g\in G\mid g+A+B=A+B\}$. Here we give a short proof of this based on a simple intersection union argument.

3 pages