  Fix a module M over a local ring R and a group action G on M, not necessarily
R-linear. To understand how large is the G-orbit of an element z\in M one looks
for the large submodules of M lying in Gz. We provide the corresponding
(necessary/sufficient) conditions in terms of the tangent space to the orbit,
T_{(Gz,z)}.
  This question originates from the classical finite determinacy problem of
Singularity Theory. Our treatment is rather general, in particular we extend
the classical criteria of Mather (and many others) to a broad class of rings,
modules and group actions.
  When a particular `deformation space' is prescribed, \Sigma\subseteq M, the
determinacy question is translated into the properties of the tangent spaces,
T_{(Gz,z)}, T_{(\Si,z)}, and in particular to the annihilator of their
quotient.
