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A note on the number of distinct elements and zero-sum subsequence lengths in cyclic groups

Claudiu Pop, George C. Ţurcaş

TL;DR

We study the relation between the support size of a zero-sum sequence in a cyclic group and the minimal zero-sum subsequence length, via the invariant $MZ(S)$. The main result shows that for a sequence $S$ in $\mathbb{Z}_n$ with $MZ(S)=n-s$ (where $s\in\{0,\dots, n-1\}$), the support satisfies $\mathrm{supp}(S)\le s+1$, with the proof leveraging the sum-set $\Sigma(S)$ and two key lemmas to derive lower bounds on its size. An inclusion-exclusion argument then yields the bound, and the paper discusses when equality can occur and its structural implications. A concrete number-theoretic application to number fields with cyclic class groups demonstrates how these combinatorial results constrain factorizations of ideals into irreducibles, connecting zero-sum theory to arithmetic of rings of integers.

Abstract

In this short note we investigate zero-sum sequences in finite abelian groups, examining the relationship between the sequence's support size, that is the number of distinct elements, and its properties concerning zero-sums. In particular, for sequences $S$ in a cyclic group, we establish a direct connection between $MZ(S)$, the length of the shortest nonempty subsequence summing to zero and the number of distinct values in $S$. Our results reveal that sequences with larger support must contain shorter non-empty zero-sum subsequences, in line with classical zero-sum results. Additionally, we present one application of our main result to a factorization of ideals problem in rings of integers of a number field.

A note on the number of distinct elements and zero-sum subsequence lengths in cyclic groups

TL;DR

We study the relation between the support size of a zero-sum sequence in a cyclic group and the minimal zero-sum subsequence length, via the invariant . The main result shows that for a sequence in with (where ), the support satisfies , with the proof leveraging the sum-set and two key lemmas to derive lower bounds on its size. An inclusion-exclusion argument then yields the bound, and the paper discusses when equality can occur and its structural implications. A concrete number-theoretic application to number fields with cyclic class groups demonstrates how these combinatorial results constrain factorizations of ideals into irreducibles, connecting zero-sum theory to arithmetic of rings of integers.

Abstract

In this short note we investigate zero-sum sequences in finite abelian groups, examining the relationship between the sequence's support size, that is the number of distinct elements, and its properties concerning zero-sums. In particular, for sequences in a cyclic group, we establish a direct connection between , the length of the shortest nonempty subsequence summing to zero and the number of distinct values in . Our results reveal that sequences with larger support must contain shorter non-empty zero-sum subsequences, in line with classical zero-sum results. Additionally, we present one application of our main result to a factorization of ideals problem in rings of integers of a number field.
Paper Structure (5 sections, 12 theorems, 18 equations)

This paper contains 5 sections, 12 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Primary results
  3. Main result
  4. Constraints in achieving the upper bound
  5. A number theoretic application

Key Result

Proposition 1.1

Let $G$ be a cyclic group of order $n$ and $(g_i)_{i=1}^{n}$ a zero-sum sequence of length $n$ with elements from $G$, which does not have a proper zero-sum subsequence. Then $g_1=...=g_{n}$.

Theorems & Definitions (34)

  • Definition 1
  • Definition 2
  • Proposition 1.1
  • proof
  • Definition 3
  • Remark
  • Example
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • ...and 24 more