  An $H$-decomposition of a graph $\Gamma$ is a partition of its edge set into
subgraphs isomorphic to $H$. A transitive decomposition is a special kind of
$H$-decomposition that is highly symmetrical in the sense that the subgraphs
(copies of $H$) are preserved and transitively permuted by a group of
automorphisms of $\Gamma$. This paper concerns transitive $H$-decompositions of
the graph $K_n \Box K_n$ where $H$ is a path. When $n$ is an odd prime, we
present a construction for a transitive path decomposition where the paths in
the decomposition are arbitrary large.
