Rainbow paths in directed graphs

An old problem in combinatorial group theory asks, given a group $Γ$ and a subset $S \subseteq Γ$, when does there exist an ordering $s_1, \cdots, s_k$ of the elements of $S$ such that the partial products $\prod_{i = 1}^j s_i$, $1 \leq j \leq k$, are all distinct. If such an ordering exists, we cal