  For a finite (not necessarily Abelian) group $(\Gamma,\cdot)$, let $n(\Gamma)
\in \mathbb{N}$ denote the smallest positive integer $n$ such that for every
labelling of the arcs of the complete digraph of order $n$ using elements from
$\Gamma$, there exists a directed cycle such that the arc-labels along the
cycle multiply to the identity. Alon and Krivelevich initiated the study of the
parameter $n(\cdot)$ on cyclic groups and proved $n(\mathbb{Z}_q)=O(q \log q)$.
This was later improved to a linear bound of $n(\Gamma)\le 8|\Gamma|$ for every
finite Abelian group by M\'{e}sz\'{a}ros and the last author, and then further
to $n(\Gamma)\le 2|\Gamma|-1$ for every non-trivial finite group independently
by Berendsohn, Boyadzhiyska and Kozma as well as by Akrami, Alon, Chaudhury,
Garg, Mehlhorn and Mehta.
  In this series of two papers we conclude this line of research by proving
that $n(\Gamma)\le |\Gamma|+1$ for every finite group $(\Gamma,\cdot)$, which
is the best possible such bound in terms of the group order and precisely
determines the value of $n(\Gamma)$ for all cyclic groups as
$n(\mathbb{Z}_q)=q+1$.
  In the present paper we prove the above result for all groups of odd order.
The proof for groups of even order needs to overcome substantial additional
obstacles and will be presented in the second part of this series.
