On tiny zero-sum sequences over finite abelian groups
Volume 168 / 2022
Abstract
Let $G$ be an additive finite abelian group. Let $S = g_1 \cdot \ldots \cdot g_l$ be a sequence over $G$, and $\mathsf k (S) = \mathrm {ord} (g_1)^{-1} + \cdots + \mathrm {ord} (g_l)^{-1}$ be its cross number. Let $\mathsf t(G)$ (resp. $\eta (G)$) be the smallest integer $t$ such that every sequence of $t$ elements, repetition allowed, from $G$ has a non-empty zero-sum subsequence $T$ with $\mathsf k(T)\leq 1$ (resp. $|T|\leq \exp (G)$). It is easy to see that $\mathsf t(G)\geq \eta (G)$. It is known that $\mathsf t(G)= \eta (G)=|G|$ when $G$ is cyclic, and for any integer $r\geq 3$ there are infinitely many groups $G$ of rank $r$ such that $\mathsf t(G) \gt \eta (G)$. Girard (2012) conjectured that $\mathsf t(G)= \eta (G)$ for all finite abelian groups of rank 2. This conjecture has been verified only for the groups $G\cong C_{p^{\alpha }}\oplus C_{p^{\alpha }}$, $G\cong C_2\oplus C_{2p}$ and $G\cong C_3\oplus C_{3p}$ with $p\geq 5$, where $p$ is a prime. We confirm this conjecture for more groups, including the groups $G\cong C_n\oplus C_n$ with the smallest prime divisor of $n$ not less than the number of distinct prime divisors of $n$.
