Minimal zero-sum sequence of length five over finite cyclic groups of prime power order

Let $G$ be a finite cyclic group. Every sequence $S$ of length $l$ over $G$ can be written in the form $S=(x_1g)cdotldotscdot(x_lg)$ where $gin G$ and $x_1, ldots, x_lin[1, ord(g)]$, and the index $ind(S)$ of $S$ is defined to be the minimum of $(x_1+cdots+x_l)/ord(g)$ over all possible $gin G$ such that $langle g rangle =G$. Recently the second and the third authors determined the index of any minimal zero-sum sequence $S$ of length 5 over a cyclic group of a prime order where $S=g^2(x_2g)(x_3g)(x_4g)$. In this paper, we determine the index of any minimal zero-sum sequence $S$ of length 5 over a cyclic group of a prime power order. It is shown that if $G=langle grangle$ is a cyclic group of prime power order $n=p^mu$ with $p geq 7$ and $mugeq 2$, and $S=(x_1g)(x_2g)(x_2g)(x_3g)(x_4g)$ with $x_1=x_2$ is a minimal zero-sum sequence with $gcd(n,x_1,x_2,x_3,x_4,x_5)=1$, then $ind(S)=2$ if and only if $S=(mg)(mg)(mfrac{n-1}{2}g)(mfrac{n+3}{2}g)(m(n-3)g)$ where $m$ is a positive integer such that $gcd(m,n)=1$.

An upper bound for Davenport constant of finite groups

Let $G$ be a finite (not necessarily abelian) group and let $p=p(G)$ be the smallest prime number dividing $|G|$. We prove that $d(G)leq frac{|G|}{p}+9p^2-10p$, where $d(G)$ denotes the small Davenport constant of $G$ which is defined as the maximal integer $ell$ such that there is a sequence over $G$ of length $ell$ contains no nonempty one-product subsequence.

Minimal zero-sum sequences of length four over finite cyclic groups II

Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)cdotldotscdot(n_lg)$ where $gin G$ and $n_1, ldots, n_lin[1, ord(g)]$, and the index $ind(S)$ of $S$ is defined to be the minimum of $(n_1+cdots+n_l)/ord(g)$ over all possible $gin G$ such that $langle g rangle =G$. An open problem on the index of length four sequences asks whether or not every minimal zero-sum sequence of length 4 over a finite cyclic group $G$ with $gcd(|G|, 6)=1$ has index 1. In this paper, we show that if $G=langle grangle$ is a cyclic group with order of a product of two prime powers and $gcd(|G|, 6)=1$, then every minimal zero-sum sequence $S$ of the form $S=(g)(n_2g)(n_3g)(n_4g)$ has index 1. In particular, our result confirms that the above problem has an affirmative answer when the order of $G$ is a product of two different prime numbers or a prime power, extending a recent result by the first author, Plyley, Yuan and Zeng.

Minimal zero-sum sequences of length five over finite cyclic groups

Let $G$ be a finite cyclic group. Every sequence $S$ of length $l$ over $G$ can be written in the form $S=(n_1g)cdotldotscdot(n_lg)$ where $gin G$ and $n_1, ldots, n_lin[1, ord(g)]$, and the index $ind(S)$ of $S$ is defined to be the minimum of $(n_1+cdots+n_l)/ord(g)$ over all possible $gin G$ such that $langle g rangle =G$. In this paper, we determine the index of any minimal zero-sum sequence $S$ of length 5 when $G=langle grangle$ 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 $G=langle grangle$ 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 $S=g^2(frac{p-1}{2}g)(frac{p+3}{2}g)((p-3)g)$.

On the Index of Sequences over Cyclic Groups

Let $G$ be a finite cyclic group of order $n ge 2$. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)cdot ... cdot (n_lg)$ where $gin G$ and $n_1,..., n_l in [1,ord(g)]$, and the index $ind (S)$ of $S$ is defined as the minimum of $(n_1+ ... + n_l)/ord (g)$ over all $g in G$ with $ord (g) = n$. In this paper we prove that a sequence $S$ over $G$ of length $|S| = n$ having an element with multiplicity at least $frac{n}{2}$ has a subsequence $T$ with $ind (T) = 1$, and if the group order $n$ is a prime, then the assumption on the multiplicity can be relaxed to $frac{n-2}{10}$. On the other hand, if $n=4k+2$ with $k ge 5$, we provide an example of a sequence $S$ having length $|S| > n$ and an element with multiplicity $frac{n}{2}-1$ which has no subsequence $T$ with $ind (T) = 1$. This disproves a conjecture given twenty years ago by Lemke and Kleitman.