  In connection to the Fuglede conjecture, we study groups of local
translations associated to spectral sets, i.e., measurable sets in $\br$ or
$\bz$ that have an orthogonal basis of exponential functions. We investigate
the connections between the groups of local translations on $\bz$ and on $\br$
and present some examples for low cardinality. We present some relations
between the group of local translations and tilings.
