  Let $G$ be a locally compact group. We examine the problem of determining
when nonzero functions in $L^2(G)$ have linearly independent translations. In
particular, we establish some results for the case when $G$ has an irreducible,
square integrable, unitary representation. We apply these results to the
special cases of the affine group, the shearlet group and the Weyl-Heisenberg
group. We also investigate the case when $G$ has an abelian, closed subgroup of
finite index.
