  We investigate under which conditions the cosmological constant vanishes
perturbatively at the one-loop level for heterotic strings on
non-supersymmetric toroidal orbifolds. To obtain model-independent results,
which do not rely on the gauge embedding details, we require that the
right-moving fermionic partition function vanishes identically in every
orbifold sector. This means that each sector preserves at least one, but not
always the same Killing spinor. The existence of such Killing spinors is
related to the representation theory of finite groups, i.e. of the point group
that underlies the orbifold. However, by going through all inequivalent
(Abelian and non-Abelian) point groups of six-dimensional toroidal orbifolds we
show that this is never possible: For any non-supersymmetric orbifold there is
always (at least) one sector, that does not admit any Killing spinor. The
underlying mathematical reason for this no-go result is formulated in a
conjecture, which we have tested by going through an even larger number of
finite groups. This conjecture could be applied to situations beyond symmetric
toroidal orbifolds, like asymmetric orbifolds.
