  We determine the index of symmetry of 3-dimensional unimodular Lie groups
with a left-invariant metric. In particular, we prove that every 3-dimensional
unimodular Lie group admits a left-invariant metric with positive index of
symmetry. We also study the geometry of the quotients by the so-called
foliation of symmetry, and we explain in what cases the group fibers over a
2-dimensional space of constant curvature.
