Inverse Zero-Sum Problems: The Property C and Property D Conjectures

Let $C_n^r$ denote the direct sum of $r$ cyclic groups of order $n$, where $n\ge 2$ and $r\ge 1$. Properties C and D predict a uniform multiplicity structure for extremal zero-sum problems over $C_n^r$: every sequence of maximum length avoiding nonempty zero-sum subsequences of length at most $n$, or of length exactly $n$, respectively, should have the form $T^{n-1}$. We give negative answers to the general Property C problem posed by Gao, Hou, Schmid and Thangadurai \cite[Open Problems 1.2]{GHST07} in 2007, and disprove the Property D conjecture proposed by Gao and Geroldinger \cite[Conjecture 7.2]{GG06} in 2006.