  Let $A$ be a multiset with elements in an abelian group. Let $FS(A)$ be the
multiset containing the $2^{|A|}$ sums of all subsets of $A$.
  We study the reconstruction problem ``Given $FS(A)$, is it possible to
identify $A$?'', and we give a satisfactory answer for all abelian groups. We
prove that, up to identifying multisets through a natural equivalence relation,
the function $A \mapsto FS(A)$ is injective (and thus the reconstruction
problem is solvable) if and only if every order $n$ of a torsion element of the
abelian group satisfies a certain number-theoretical property linked to the
multiplicative group $(\mathbb{Z} / n\mathbb{Z})^*$.
  The core of the proof relies on a delicate study of the structure of
cyclotomic units. Moreover, as a tool, we develop an inversion formula for a
novel discrete Radon transform on finite abelian groups that might be of
independent interest.
