  We consider the problem of uniqueness of certain simultaneity structures in
flat spacetime. Absolute simultaneity is specified to be a non-trivial
equivalence relation which is invariant under the automorphism group Aut of
spacetime. Aut is taken to be the identity-component of either the
inhomogeneous Galilei group or the inhomogeneous Lorentz group. Uniqueness of
standard simultaneity in the first, and absence of any absolute simultaneity in
the second case are demonstrated and related to certain group theoretic
properties. Relative simultaneity with respect to an additional structure X on
spacetime is specified to be a non-trivial equivalence relation which is
invariant under the subgroup in Aut that stabilises X. Uniqueness of standard
Einstein simultaneity is proven in the Lorentzian case when X is an inertial
frame. We end by discussing the relation to previous work of others.
