Spanning subset sums for finite Abelian groups Original Research Article

We survey the state of research to determine the maximum size of a nonspanning subset of a finite abelian group G of order n. The smallest prime factor of n, denote it here by p, plays a crucial role. For prime order, G=Zp, this is essentially an old problem of Erdős and Heilbronn, which can be solved using a result of Dias da Silva and Hamidoune. We provide a simple new proof for the solution when n is even (p=2). For composite odd n, we deduce the solution, for n⩾2p2, from results obtained years ago by Diderrich and, recently, by Gao and Hamidoune. Only a small family of cases remains unsettled.