Extra invariance of principal shift invariant spaces and the Zak transform.

We prove a necessary and sufficient condition for a principal shift invariant space of $L^2(mathbb{R})$ to be invariant under translations by the subgroup $frac{1}{N} mathbb{Z}, Nu003e1$. This condition is given in terms of the Zak transform of the group $frac{1}{N} mathbb{Z}.$ This result is extended to principal shift invariant spaces generated by a lattice in a general locally compact abelian (LCA) group.