  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.
