  The germs of maps (k^n,o)\to(k^p,o) are traditionally studied up to the
right, left-right or contact equivalence. Various questions about the
group-orbits are reduced to their tangent spaces. Classically the passage from
the tangent spaces to the orbits was done by vector fields integration, hence
it was bound to the real/complex-analytic or C^r-category.
  The purely-algebraic (characteristic-free) approach to the group-orbits of
right and contact equivalence has been developed during the last decades. But
those methods could not address the (essentially more complicated) left-right
equivalence. Moreover, the characteristic-free results (in the right/contact
cases) were weaker than those in characteristic zero, because of the
(inevitable) pathologies of positive characteristic.
  In this paper we close these omissions.
  * We establish the general (characteristic-free) passage from the tangent
spaces to the groups orbits for the groups of right, contact and let-right
equivalence. Submodules of the tangent spaces ensure (shifted) submodules of
the group-orbits.
  For the left-right equivalence this extends (and strengthens) various
classical results of Mather, Gaffney, du Plessis, and others.
  * A filtration on the space of maps induces the filtration on the group and
on the tangent space.
  We establish the criteria of type "$T_{G^{(j)}}f$ vs $G^{(j)} f$" in their
strongest form, for arbitrary base field/ring, provided the characteristic is
zero or high for a given map. This brings the "inevitably weaker" results of
char>0 to the level of char=0.
  * As an auxiliary step, important on its own, we develop the mixed-module
structure of the tangent space to the left-right group and establish various
properties of the annihilator ideal (that defines the instability locus of the
map).
