  Given a non-trivial finite Abelian group $(A,+)$, let $n(A) \ge 2$ be the
smallest integer such that for every labelling of the arcs of the bidirected
complete graph of order $n(A)$ with elements from $A$ there exists a directed
cycle for which the sum of the arc-labels is zero. The problem of determining
$n(\mathbb{Z}_q)$ for integers $q \ge 2$ was recently considered by Alon and
Krivelevich, who proved that $n(\mathbb{Z}_q)=O(q \log q)$. Here we improve
their bound and show that $n(\mathbb{Z}_q)$ grows linearly. More generally we
prove that for every finite Abelian group $A$ we have $n(A) \le 8|A|$, while if
$|A|$ is prime then $n(A) \le \frac{3}{2}|A|$. As a corollary we also obtain
that every $K_{16q}$-minor contains a cycle of length divisible by $q$ for
every integer $q \ge 2$, which improves a result by Alon and Krivelevich.
