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Finite determinacy for mappings has been classically thoroughly studied innumerous scenarios in the real- and complex-analytic category and in thedifferentiable case. It means that the map-germ is determined, up to a givenequivalence relation, by a finite part of its Taylor expansion. The equivalencerelation is usually given by a group action and the first step is always toreduce the determinacy question to an "infinitesimal determinacy", i.e., to thetangent spaces at the orbits of the group action. In this work we formulate auniversal, characteristic-free approach to finite determinacy, not necessarilyover a field, and for a large class of group actions. We do not restrict topro-algebraic or Lie groups, rather we introduce the notion of "pairs of (weak)Lie type", which are groups together with a substitute for the tangent space tothe orbit such that the orbit is locally approximated by its tangent space, ina precise sense. This construction may be considered as a kind of replacementof the exponential resp. logarithmic maps. It is of independent interest as itprovides a general method to pass from the tangent space to the orbit of agroup action in any characteristic. In this generality we establish the"determinacy versus infinitesimal determinacy" criteria, a far reachinggeneralization of numerous classical and recent results, together with some newapplications.

Pairs of Lie-type and large orbits of group actions on filtered modules. (A characteristic-free approach to finite determinacy.)