[Paper Review] Minimal zero-sum sequences of length five over finite cyclic groups

Jiangtao Peng, Yuanlin Li|arXiv (Cornell University)|Mar 7, 2013

Finite Group Theory Research5 references3 citations

TL;DR

This paper determines the index of minimal zero-sum sequences of length five over finite cyclic groups of prime order $ p \geq 31 $. It proves that such sequences have index 1 unless they are of the form $ S = g^2\left(\frac{p-1}{2}g\right)\left(\frac{p+3}{2}g\right)((p-3)g) $, in which case the index is 2. The result completes the classification of index values for length-5 minimal zero-sum sequences with element multiplicity at least 2.

ABSTRACT

Let $G$ be a finite cyclic group. Every sequence $S$ of length $l$ over $G$ can be written in the form $S=(n_1g)\cdot\ldots\cdot(n_lg)$ where $g\in G$ and $n_1, \ldots, n_l\in[1, \ord(g)]$, and the index $\ind(S)$ of $S$ is defined to be the minimum of $(n_1+\cdots+n_l)/\ord(g)$ over all possible $g\in G$ such that $\langle g angle =G$. In this paper, we determine the index of any minimal zero-sum sequence $S$ of length 5 when $G=\langle g angle$ is a cyclic group of a prime order and $S$ has the form $S=g^2(n_2g)(n_3g)(n_4g)$. It is shown that if $G=\langle g angle$ is a cyclic group of prime order $p \geq 31$, then every minimal zero-sum sequence $S$ of the above mentioned form has index 1 except in the case that $S=g^2(\frac{p-1}{2}g)(\frac{p+3}{2}g)((p-3)g)$.

Motivation & Objective

• To determine the index of minimal zero-sum sequences of length 5 over finite cyclic groups of prime order $ p \geq 31 $, particularly when the maximum element repetition is at least 2.
• To resolve the open problem of characterizing which such sequences have index 2, building on prior results for index 1 sequences.
• To provide a complete classification of index values for minimal zero-sum sequences of length 5 with $ \mathsf{h}(S) \geq 2 $, complementing earlier results on higher or lower multiplicities.
• To establish a precise condition under which the index is 2, thereby closing a key gap in the understanding of index behavior in zero-sum sequence theory.

Proposed method

• The authors define the index of a sequence $ S = (n_1 g) \cdots (n_l g) $ as $ \operatorname{ind}(S) = \min \left\{ \frac{n_1 + \cdots + n_l}{\operatorname{ord}(g)} \right\} $ over all generators $ g $ of the group $ G $.
• They analyze sequences of the form $ S = g^2 (n_2 g)(n_3 g)(n_4 g) $ with $ n_i \in [1, p-1] $, focusing on zero-sum and minimality conditions.
• The proof uses modular arithmetic and case analysis based on the values of $ n_2, n_3, n_4 $, reducing to analyzing the least positive residue modulo $ p $.
• They apply lemmas on bounds of ratios involving $ p $, $ c = n_2 + n_3 + n_4 - 2 $, and $ b = n_2 + n_3 $, using inequalities to eliminate configurations with index 2.
• The analysis distinguishes cases based on the value of $ a = \max(n_2, n_3, n_4) $, particularly $ a = \ell + 1 $ and $ a = \ell + 2 $, and uses bounds on $ p/c $ and $ p/b $ to derive contradictions unless in the exceptional case.
• They use known results (Lemmas 2.3–2.4, 3.1–3.6) to rule out index 2 for all but one specific sequence, confirming that only $ S = g^2\left(\frac{p-1}{2}g\right)\left(\frac{p+3}{2}g\right)((p-3)g) $ has index 2.

Experimental results

Research questions

• RQ1 For a minimal zero-sum sequence $ S $ of length 5 over a cyclic group $ G $ of prime order $ p \geq 31 $, when is the index $ \operatorname{ind}(S) = 2 $?
• RQ2 What is the complete set of sequences of length 5 with $ \mathsf{h}(S) \geq 2 $ and index 2 over $ \mathbb{Z}_p $?
• RQ3 Can the index of all minimal zero-sum sequences of length 5 with $ \mathsf{h}(S) \geq 2 $ be fully classified over $ \mathbb{Z}_p $?
• RQ4 Is there a unique exceptional sequence of length 5 with index 2, and if so, what is its exact form?

Key findings

• For a cyclic group $ G = \langle g \rangle $ of prime order $ p \geq 31 $, every minimal zero-sum sequence $ S $ of length 5 with $ \mathsf{h}(S) \geq 2 $ has index either 1 or 2.
• The index of $ S $ is 2 if and only if $ S = g^2\left(\frac{p-1}{2}g\right)\left(\frac{p+3}{2}g\right)((p-3)g) $, and this is the only such sequence with index 2.
• All other minimal zero-sum sequences of length 5 with $ \mathsf{h}(S) \geq 2 $ over $ \mathbb{Z}_p $ have index 1.
• The result completes the classification of index values for minimal zero-sum sequences of length 5 with $ \mathsf{h}(S) \geq 2 $, as the case $ \mathsf{h}(S) = 1 $ remains open.