The subsums of zero-sum free sequences in finite cyclic groups

Abstract

Let ${\mathbb{Z}}_n$ be the cyclic group of order $n\ge 3$ additively written. S. Savchev \& F. Chen (2007) proved that for each zero-sum free sequence $S=a_1\bullet \cdots \bullet a_t$ over ${\mathbb{Z}}_n$ of length $t>n/2$, there is an integer $g$ coprime to $n$ such that, if $\overline{r}$ denotes the least positive integer in the congruence class $r$ modulo $n$, then ${\sum }_{i=1}^t\overline{g{a}_i}<n$. Under the same hypothesis, in this paper we show that $$\left\{\sum _{i\in \Lambda }\overline{g{a}_i}|\Lambda \subset \{1,2,\dots ,t\}\right\}=\left\{1,2,\dots ,\sum _{i=1}^t\overline{g{a}_i}\right\}.$$ It simplifies many calculations on inverse zero-sum problems.

7 pages