  The motivation for this paper is to extend the known model theoretic
treatment of differential Galois theory to the case of linear difference
equations (where the derivative is replaced by an automorphism.) The model
theoretic difficulties in this case arise from the fact that the corresponding
theory ACFA does not eliminate quantifiers. We therefore study groups of
restricted automorphisms, preserving only part of the structure. We give
conditions for such a group to be (infinitely) definable, and when these
conditions are satisfied we describe the definition of the group and the action
explicitly.
  We then examine the special case when the theory in question is obtained by
enriching a stable theory with a generic automorphism. Finally, we interpret
the results in the case of ACFA, and explain the connection of our construction
with the algebraic theory of Picard-Vessiot extensions.
  The only model theoretic background assumed is the notion of a definable set.
