Some extensions of the Cauchy-Davenport theorem

The Cauchy-Davenport theorem states that, if p is prime and A, B are nonempty subsets of cardinality r, s in $\mathbb{Z}/p\mathbb{Z}$, the cardinality of the sumset $A+B=\{a+b|a\in A,b\in B\}$ is bounded below by $\min (r+s-1,p)$; moreover, this lower bound is sharp. Natural extensions of this result consist in determining, for each group G and positive integers $r,s⩽|G|$, the analogous sharp lower bound, namely the function${\mu }_G(r,s)=\min \{|A+B||A,B\subset G,|A|=r,|B|=s\}.$ Important progress on this topic has been achieved in recent years, leading to the determination of ${\mu }_G$ for all abelian groups G. In this note we survey the history of earlier results and the current knowledge on this function.