The Exact Minimum Order for Unique Multiset Sums in Finite Abelian Groups

This project presents a proof of the exact minimum order for unique multiset sums in finite abelian groups, resolving an open problem posed by J. A. R. Fonollosa. It establishes the sharp bound 2^{n-1}, classifies all equality cases, and derives a consequence for finite-abelian character schemes used in permanent evaluation.