On the Index of Sequences over Cyclic Groups

TL;DR

This paper investigates the index of sequences over cyclic groups, proving conditions under which a subsequence with index 1 exists, and providing counterexamples that disprove a long-standing conjecture.

Contribution

It establishes new conditions for the existence of subsequences with index 1 and disproves a conjecture by Lemke and Kleitman through counterexamples.

Findings

Sequences with high element multiplicity contain subsequences with index 1.

For prime order groups, the multiplicity condition can be relaxed.

Counterexamples exist for certain group orders, disproving the conjecture.

Abstract

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 $g\in G$ 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.