Extra invariance of principal shift invariant spaces and the Zak transform

Abstract: 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 $\frac{1}{N} \mathbb{Z}, N>1$. 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.