Zero-sum invariants of finite abelian groups

TL;DR

This paper introduces a unified framework for zero-sum invariants in finite abelian groups, defining key parameters and presenting initial results and open problems in the area.

Contribution

It formulates a general approach to zero-sum invariants and provides initial findings and open questions for the invariant $d_{ ext{Omega}}(G)$.

Findings

Initial results on the invariant $d_{ ext{Omega}}(G)$

Open problems proposed for future research

Framework unifying various zero-sum invariants

Abstract

The purpose of the article is to provide an unified way to formulate zero-sum invariants. Let $G$ be a finite additive abelian group. Let $B(G)$ denote the set consisting of all nonempty zero-sum sequences over G. For $\Omega \subset B(G$), let $d_{\Omega}(G)$ be the smallest integer $t$ such that every sequence $S$ over $G$ of length $|S|\geq t$ has a subsequence in $\Omega$.We provide some first results and open problems on $d_{\Omega}(G)$.

Equations

$$d_{\Omega}(G)=\left\{\begin{array}[]{ll}\mathsf{D}(G),\mbox{ if }\Omega=\mathcal{B}(G);\\ \eta(G),\mbox{ if }\Omega=\{S\in\mathcal{B}(G):1\leq|S|\leq\exp(G)\};\\ \mathsf{s}(G),\mbox{ if }\Omega=\{S\in\mathcal{B}(G):|S|=\exp(G)\};\\ \mathsf{E}(G),\mbox{ if }\Omega=\{S\in\mathcal{B}(G):|S|=|G|)\};\\ \mathbf{d}_{\mathcal{L}}(G),\mbox{ if }\Omega=\{S\in\mathcal{B}(G):|S|\in\mathcal{L}\}.\end{array}\right.$$

$$G\cong C_{n_1}\oplus \cdots \oplus C_{n_r},$$

$$M(G)=1+\sum _{i=1}^r(n_i-1).$$

$$S=\prod _{g\in G}{g}^{{\text{v}}_g(S)}\text{ with }{\text{v}}_g(S)\in {\mathbb{N}}_0\text{ for all }g\in G_0.$$

$$\mathsf{D}(G)\in Vol(G),\mathsf{D}(G)+1\in Vol(G)\text{mbox}and\eta (G)\in Vol(G).$$

$$q^{\prime }(G)\le disc(G)\le {\mathsf{D}}_2(G)$$

$$\Omega \cap {\Omega }^{\prime }=\varnothing .$$

Full text

1. Introduction

Zero-sum theory on abelian groups can be traced back to 1960’s and has been developed rapidly in recent three decades (see [2], [7] and [8]). Many invariants have been formulated in zero-sum theory and we list some of these invariants here. Let $G$ be an additive finite abelian group. The EGZ-constant $\mathsf{s}(G)$ is the smallest integer $t$ such that every sequence $S$ over $G$ of length $|S|\geq t$ has a zero-sum subsequence of length $\exp(G)$. This invariant $\mathsf{s}(G)$ comes from the so called Erdős-Ginzburg-Ziv theorem, which asserts that $\mathsf{s}(G)\leq 2|G|-1$ and is regarded as one of two starting points of zero-sum theory. Another starting point of zero-sum theory involves the Davenport constant $\mathsf{D}(G)$, which is defined as the smallest integer $t$ such that every sequence $S$ over $G$ of length $|S|\geq t$ has a nonempty zero-sum subsequence. This invariant $\mathsf{D}(G)$ was formulated by H. Davenport in 1965 and his motivation was to study algebraic number theory. Let $\eta(G)$ be the smallest integer $t$ such that every sequence $S$ over $G$ of length $|S|\geq t$ has a zero-sum subsequence of length between 1 and $\exp(G)$, which was first introduced by Emade Boas in 1969 to study $\mathsf{D}(G)$. Let $\mathsf{E}(G)$ be the smallest integer $t$ such that every sequence $S$ over $G$ of length $|S|\geq t$ has a zero-sum subsequence of length $|G|$, which was formulated by the first author in 1996. The above invariants on zero-sum sequences have been studied by many authors. To give a unified way of formulating zero-sum invariants, recently, Geroldinger, Grynkiewicz and Schmid [6] introduced the following concept. Let $\mathcal{L}$ be a finite set of positive integers, and let $\mathbf{d}_{\mathcal{L}}(G)$ be the smallest integer $t$ such that every sequence $S$ over $G$ of length $|S|\geq t$ has a zero-sum subsequence of length in $\mathcal{L}$. Here we give a more unified way to formulate zero-sum invariants.

Let $\mathcal{B}(G)$ denote the set consisting of all nonempty zero-sum sequences over $G$. For $\Omega\subset\mathcal{B}(G)$, define $d_{\Omega}(G)$ to be the smallest integer $t$ such that every sequence $S$ over $G$ of length $|S|\geq t$ has a subsequence in $\Omega$. If such $t$ does not exist, we let $d_{\Omega}(G)=\infty$. Now we have

[TABLE]

In this paper, we will provide some first results on $d_{\Omega}(G)$ and formulate some open problems. The rest of this paper is organized as follows. In Section 2, we present some necessary concepts and terminology. In Section 3 we study the way to present a given integer $t\geq\mathsf{D}(G)$ by $\Omega\subset\mathcal{B}(G)$ with $d_{\Omega}(G)=t$; In the final section we study the minimal $\Omega\subset\mathcal{B}(G)$ with respect to $d_{\Omega}(G)=t$.

2. Preliminaries

Let $\mathbb{N}$ denote the set of positive integers, $\mathbb{N}_{0}=\mathbb{N}\cup\{0\}$. For a real number $x$, we denote by $\lfloor x\rfloor$ the largest integer that is less than or equal to $x$.

Throughout, all abelian groups will be written additively. By the Fundamental Theorem of Finite Abelian Groups we have

[TABLE]

where $r=\mathsf{r}(G)\in\mathbb{N}_{0}$ is the rank of $G$, $n_{1},\ldots,n_{r}\in\mathbb{N}$ are integers with $1<n_{1}|\ldots|n_{r}$, moreover, $n_{1},\ldots,n_{r}$ are uniquely determined by $G$, and $n_{r}=\text{exp}(G)$ is the $exponent$ of $G$. Set

[TABLE]

Let $G_{0}\subset G$. For $g_{1},\ldots,g_{l}\in G_{0}$(repetition allowed), we call $S=g_{1}\cdot\ldots\cdot g_{l}$ a $sequence$ over $G_{0}$. We write sequences $S$ in the form

[TABLE]

We call $\textsf{v}_{g}(S)$ the $multiplicity$ of $g$ in $S$.

For $S=g_{1}\cdot\ldots\cdot g_{l}=\prod_{g\in G_{0}}g^{\textsf{v}_{g}(S)}$, we call

• •

$|S|=l=\sum_{g\in G_{0}}\textsf{v}_{g}(S)\in\mathbb{N}_{0}$ the $length$ of $S$.

• •

$\sigma(S)=\sum\limits_{i=1}\limits^{l}g_{i}=\sum_{g\in G_{0}}\textsf{v}_{g}(S)g\in G_{0}$ the $sum$ of $S$.

• •

$S$ is a $zero$-$sum$ $sequence$ if $\sigma(S)=0$.

• •

$S$ is a $short$ $zero$-$sum$ $sequence$ if it is a zero-sum sequence of length $|S|\in[1,\text{exp}(G)]$

We denote by $\mathcal{B}(G_{0})$ the set of all nonempty zero-sum sequences over $G_{0}$, by $\mathcal{A}(G_{0})$ the set of all minimal zero-sum sequences over $G_{0}$.

We say two sequences $S$ and $T$ over $G$ have the same form if and only if $\textsf{v}_{g}(S)=\textsf{v}_{g}(T)$ for every $g\in G$.

3. Representing the same invariant $t$ by different $\Omega$

with $d_{\Omega}(G)=t$

We first state some basic properties on $d_{\Omega}(G)$.

Proposition 3.1**.**

Let $\Omega,\Omega^{\prime}\subset\mathcal{B}(G)$.

$d_{\Omega}(G)<\infty$* if and only if for every element $g\in G$, $g^{k\operatorname{ord}(g)}\in\Omega$ for some positive integer $k=k(g)$* 2. 2.

If $\Omega\subset\Omega^{\prime}$ then $d_{\Omega^{\prime}}(G)\leq d_{\Omega}(G).$ 3. 3.

If $S_{1}$ is a proper subsequence of $S$ and both $S$ and $S_{1}$ belong to $\Omega$, then $d_{\Omega\setminus\{S_{1}\}}(G)=d_{\Omega}(G)$. 4. 4.

For every positive integer $t\in[\mathsf{D}(G),\infty),$ there is an $\Omega\subset\mathcal{B}(G)$ such that $t=d_{\Omega}(G).$

Proof.

1. Necessity. Let $t=d_{\Omega}(G)<\infty$. For every $g\in G$, the sequence $g^{t}$ has a nonempty zero-sum subsequence $S$ in $\Omega$. Since $S$ is zero-sum, we infer that it has the form $g^{k\operatorname{ord}(g)}$ for some positive integer $k$. This proves the necessity.

Sufficiency. If for every $g\in G$ there is a positive integer $k_{g}$ such that $g^{k_{g}\operatorname{ord}(g)}\in\Omega$, then clearly $d_{\Omega}(G)\leq 1+\sum_{g\in G}(k_{g}\operatorname{ord}(g)-1)$ completing the proof of Conclusion 1.

2. The result holds obviously.

3. The result follows from 2.

4. Let $G^{*}=G\setminus\{0\}$. For each positive integer $t\geq\mathsf{D}(G)$. Let $\Omega=\{0^{t-\mathsf{D}(G)+1}\}\cup\mathcal{B}(G^{*})$. It is easy to see that $t=d_{\Omega}(G)$, completing the proof. ∎

Although Conclusion 4 of the above proposition asserts that for every positive $t\geq\mathsf{D}(G)$, there is an $\Omega\subset\mathcal{B}(G)$ such that $t=d_{\Omega}(G)$, it does not give us much information on the invariant $t$. For some classical invariants $t$, we hope to find some special $\Omega\in\mathcal{B}(G)$ with $d_{\Omega}(G)=t$ to help us understand $t$ better. We say a sequence $S$ over $G$ a weak-regular sequence if $\mathsf{v}_{g}(S)\leq\operatorname{ord}(g)$ for every $g\in G$. We say $\Omega\subset\mathcal{B}(G)$ is weak-regular if very sequence $S\in\Omega$ is weak-regular. Define $Vol(G)$ to be the set of all positive integer $t\in[\mathsf{D}(G),1+\sum_{g\in G}(\operatorname{ord}(g)-1)]$ such that $t=d_{\Omega}(G)$ for some weak-regular $\Omega\subset\mathcal{B}(G)$.

Remark 3.2**.**

A sequence over $G$ is called a regular sequence if $|S_{H}|\leq|H|-1$ for every proper subgroup $H$ of $G$, where $S_{H}$ denote the subsequence consisting of all terms of $S$ in $H$. The concept of regular sequences was introduced by Gao, Han and Zhang [3] quite recently.

Question 1. Does $Vol(G)=[\mathsf{D}(G),1+\sum_{g\in G}(\operatorname{ord}(g)-1)]$ hold for any finite abelian group $G$?

Let $\Omega\subset\mathcal{B}(G)$, we say a sequence $S$ over $G$ is $\Omega$-free if $S$ has no subsequence in $\Omega$.

Lemma 3.3**.**

Let $\Omega\in\mathcal{B}(G)$ with $d_{\Omega}(G)=t\in[\mathsf{D}(G),1+\sum_{g\in G}(\operatorname{ord}(g)-1)]$. If there is an $\Omega$-free sequence $T$ over $G$ of length $|T|=t-1$ such that $\mathsf{v}_{g}(T)\leq\operatorname{ord}(g)-1$ for every $g\in G$, then $d_{\Omega}(G)=t\in Vol(G)$.

Proof.

If every sequence in $\Omega$ is weak regular, then by the definition of $Vol(G)$ we know that $t\in Vol(G)$. Otherwise, take an $S\in\Omega$ such that $\mathsf{v}_{g}(S)\geq\operatorname{ord}(g)+1$ for some $g\in G$. Let $\Omega_{1}=(\Omega\setminus\{S\})\cup\{g^{\operatorname{ord}(g)}\}$. It is easy to see that $d_{\Omega_{1}}(G)\leq d_{\Omega}(G)=t$. On the other hand, $T$ is also $\Omega_{1}$-free. Therefore, $d_{\Omega_{1}}(G)\geq t$. Hence, $d_{\Omega_{1}}(G)=t$. Continuing the same process above we finally get an $\Omega_{k}$ which is weak regular such that $d_{\Omega_{k}}(G)=t\in Vol(G)$. ∎

Proposition 3.4**.**

For every finite abelian group $G$ we have

[TABLE]

Proof.

Let $\Omega=\mathcal{B}(G)$. Then, $d_{\Omega}(G)=\mathsf{D}(G)$. Let $T$ be a zero-sum-free sequence (or equivalently, $\Omega$-free) over $G$ of length $|T|=\mathsf{D}(G)-1=d_{\Omega}(G)-1$. Clearly, $T$ is weak regular and $\mathsf{D}(G)=d_{\Omega}(G)\in Vol(G)$ by Lemma 3.3.

To prove $\mathsf{D}(G)+1\in Vol(G)$, let $\Omega=\{S\in\mathcal{B}(G),|S|<\mathsf{D}(G)\}\cup\{g^{\operatorname{ord}(G)},g\in G\}$. It is easy to see that $d_{\Omega}(G)=\mathsf{D}(G)+1$. Let $\Omega^{\prime}=\{S\in\mathcal{A}(G),|S|<\mathsf{D}(G)\}\cup\{g^{\operatorname{ord}(G)},g\in G\}$. Clearly, $d_{\Omega^{\prime}}(G)=d_{\Omega}(G)=\mathsf{D}(G)+1$ and $\mathsf{D}(G)+1\in Vol(G)$ follows from $\Omega^{\prime}$ is weak-regular.

To prove $\eta(G)\in Vol(G)$, let $\Omega=\{S\in\mathcal{B}(G),|S|\leq\exp(G)\}$. It is easy to see that $d_{\Omega}(G)=\eta(G)$. Let $T$ be any $\Omega$-free sequence of the maximal length $|T|=\eta(G)-1$. Clearly, $\mathsf{v}_{g}(T)\leq\operatorname{ord}(g)-1$ for every $g\in G$ and hence $\eta(G)\in Vol(G)$ by Lemma 3.3 again. ∎

4. The minimal $\Omega$ with respect to $d_{\Omega}(G)=t$ for some given $t$

For a given $t\geq\mathsf{D}(G)$ and an $\Omega\subset\mathcal{B}(G)$ with $d_{\Omega}(G)=t$. We say that $\Omega$ is minimal respect to $d_{\Omega}(G)=t$ if $d_{\Omega^{\prime}}(G)>t$ for every proper subset $\Omega^{\prime}\subsetneq\Omega$.

By Proposition 3.1 (3) we have the following.

Proposition 4.1**.**

Let $t\geq\mathsf{D}(G)$ be an integer, and let $\Omega\subset\mathcal{B}(G)$ be minimal with respect to $d_{\Omega}(G)=t$. Then, for every $B,B^{\prime}\in\Omega$ with $B\neq B^{\prime}$ we have $B^{\prime}\nsubseteq B$.

Question 2. Let $\Omega\in\mathcal{B}(G)$ be minimal with respect to $d_{\Omega}(G)=\mathsf{D}(G)$. What can be said about $\Omega$?

For Question 2, $d_{\Omega}(G)=\mathsf{D}(G)$ if $\Omega=\mathcal{A}(G)$, but we can not say that $\mathcal{A}(G)$ is smallest. Let $p$ be a prime. Let $G=C_{p}$. Let $\Omega_{1}$ be the set consisting of all minimal zero-sum sequences over $G$ with index

1. If the Lemke-Kleitman’s conjecture is true for $p$, i.e., every sequence over $G$ of length $p$ has a subsequence with index 1, then $\Omega_{1}$ is a proper subset of $\mathcal{A}(G)$ with $d_{\Omega_{1}}(G)=\mathsf{D}(G)=p$. So if $n=p\geq 7$ is a prime and the Lemke-Kleitman’s conjecture mentioned above is true for $p$, then $\Omega=\mathcal{A}(G)$ is not minimal with respect to $d_{\Omega}(G)=\mathsf{D}(G)=p$. In fact, except for some small primes $p$, we don’t know any minimal $\Omega\subset\mathcal{B}(G)$ with respect to $d_{\Omega}(G)=\mathsf{D}(G)=p$. For sufficiently large $t$, we can find minimal $\Omega$ with respect to $d_{\Omega}(G)=t$.

Proposition 4.2**.**

Let $G$ be a finite abelian group and $t\geq 1+\sum_{g\in G}(\operatorname{ord}(g)-1)$ be a integer, and let $\Omega=\{g^{\operatorname{ord}(g)},g\in G\}\cup\{0^{t-\sum_{g\in G}(\operatorname{ord}(g)-1)}\}$. Then $\Omega$ is minimal with respect to $d_{\Omega}(G)=t$.

Proof.

Clearly.

∎

For a given $t\in[\mathsf{D}(G),\sum_{g\in G}(\operatorname{ord}(g)-1)]$ not too close to $\sum_{g\in G}(\operatorname{ord}(g)-1)$, it could be very difficult to find a minimal $\Omega\subset\mathcal{B}(G)$ with respect to $d_{\Omega}(G)=t.$ We provide one more example here:

Proposition 4.3**.**

Let $n\geq 5$ be an odd integer. Then $\Omega=\{T\in\mathcal{B}(C_{n})|\mbox{ }|T|=n\}$ is not minimal with respect to $d_{\Omega}(G)=2n-1$.

Proof.

Let $S_{0}=\prod_{i=0}^{n-1}i$. Since $n$ is odd, we have $S_{0}\in\Omega$. Let $\Omega_{1}=\Omega\setminus\{S_{0}\}$. Clearly, $d_{\Omega_{1}}(G)\geq d_{\Omega}(G)=2n-1$. So, it suffices to prove $d_{\Omega_{1}}(G)\leq 2n-1.$ Let $S$ be an arbitrary sequence over $G$ with length $|S|=2n-1$. We need to show $S$ contains a zero-sum subsequence in $\Omega_{1}$. Since $d_{\Omega}(G)=2n-1$, we may assume that all zero-sum subsequences of $S$ with length $n$ is of the same form as $S_{0}=\prod_{i=0}^{n-1}i$. It follows that $j$ occurs exactly once in $S$ for some $j\in[0,n-1]$ . Consider the sequence $S_{1}=S\setminus\{j\}$. Then, $|S_{1}|=2n-2$ and $S_{1}$ contains $n-1>2$ distinct elements. By recalling the well known fact that a sequence over $C_{n}$ with length $2n-2$ having no zero-sum subsequence of length $n$ if and only if the sequence consists of two distinct elements with each appearing $n-1$ times, we derive that $S_{1}$ and hence $S$ has a zero-sum subsequence in $\Omega_{1}$. ∎

We say a zero-sum sequence $S$ is essential with respect to some $t\geq\mathsf{D}(G)$ if every $\Omega\in B(G)$ with $d_{\Omega}(G)=t$ contains $S$. For examples, for $G=C_{n}$ and $t=n$, the zero-sum sequence $g(-g)$ is essential with respect to $n$ for every generator $g$ of $C_{n}$. For any finite abelian group $G$, every minimal zero-sum sequence over $G$ of length $\mathsf{D}(G)$ is essential with respect to $\mathsf{D}(G)$.

A natural research problem is to determine the smallest integer $t$ such that there is no essential zero-sum sequence with respect to $t$, denote it by $q(G)$.

Let $q^{\prime}(G)$ be the smallest integer $t$ such that, every sequence $S$ over $G$ of length $|S|\geq t$ has two nonempty zero-sum subsequences with different forms.

Question 3. Does $q(G)=q^{\prime}(G)$ for any finite abelian group $G$?

Relating to $q^{\prime}(G)$ we present two invariants below.

Let $disc(G)$ denote the smallest integer $t$ such that every sequence $S$ over $G$ of length $|S|\geq t$ has two nonempty zero-sum subsequences with different lengths. The invariants $disc(G)$ was formulated by Girard [9] and has been studied recently in [4] and [5].

Let $\mathsf{D}_{2}(G)$ denote the smallest integer $t$ such that every sequence over $G$ has two disjoint nonempty zero-sum subsequences. Clearly,

[TABLE]

for every finite abelian group $G$.

$\mathsf{D}_{2}(G)$ was first introduced by H. Halter-Koch in [10] and was studied recently by Plagne and Schmid in [11].

We close this section with the following theorem.

Theorem 4.4**.**

If $G$ is a finite abelian group then $q(G)\leq\mathsf{D}_{2}(G)$.

Proof.

Let $G^{*}=G\setminus\{0\}$. Let $k\geq\mathsf{D}_{2}(G)$ be a integer and we only need to show that there exist two disjoint subsets $\Omega,\Omega^{\prime}\subset\mathcal{B}(G)$ such that $d_{\Omega}(G)=k=d_{\Omega^{\prime}}(G)$. Let $\Omega=\{0^{k-\mathsf{D}(G)+1}\}\cup\mathcal{A}(G^{*})$ and let $\Omega^{\prime}=\{0^{k-\mathsf{D}_{2}(G)+1}\}\cup\{\mathcal{B}(G^{*})\setminus\mathcal{A}(G^{*})\}$.

Note that a minimal zero-sum sequence over $G$ of length $\mathsf{D}(G)$ has no two disjoint nonempty zero-sum subsequences, so we obtain that $\mathsf{D}_{2}(G)>\mathsf{D}(G)$. Thus,

[TABLE]

On the other hand, it is easy to see that $d_{\Omega}(G)=k=d_{\Omega^{\prime}}(G)$ and we are done.

∎

Acknowledgements. This work was supported by the PCSIRT Project of the Ministry of Science and Technology, the National Science Foundation of China, and a Discovery Grant from the Natural Science and Engineering Research Council of Canada.