The simple ordering conjecture for zero-sum subsets of abelian groups

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. The simple ordering conjecture for zero-sum subsets of abelian groups

   Let $(G,+)$ be an abelian group. Let $A$ be a finite subset of $G\setminus\{0\}$ such that no $2$-subset $\{x,-x\}$ is contained in $A$ and

   $$\sum_{a\in A}a=0.$$

   A simple ordering of $A$ is an ordering of its elements whose successive partial sums are distinct. Simple ordering conjecture. There exists a simple ordering of the elements of $A$. This conjecture concerns sequencing zero-sum subsets of abelian groups under the condition that no element and its inverse both occur; its status is not established by the supplied text.

   source: Simone Costa, Fiorenza Morini, Anita Pasotti and Marco Antonio Pellegrini, “A generalization of Heffter arrays”, arXiv:1906.03932 (2019).