  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.
