This is joint work with Ehud Hrushovski

A difference field is a field with a distinguished automorphism, s. We will work in a large existentially closed difference field U. I will discuss two notions of internality to the fixed field F (={x : s(x)=x}), and why we chose the one we did. This notion gives rise to a notion of binding group: If our internal set is defined over the difference field K, and Q is its set of solutions in U, then we consider the difference field L

generated by Q over K, and G=Aut_s(L/KF).

This group, as well as its action on Q, are definable in U, maybe with additional parameters. We show that it is isomorphic to H(F), where H is an algebraic group, and that the set Q is a translation variety, i.e., there is an algebraic set X on which H acts, and Q is isomorphic to the

set {x\in X : s(x)=h.x} for some fixed element h in H(U).

We will also discuss under which conditions all this data is defined over the original difference field K. This is in particular the case when K is algebraically closed and F\cap K is pseudo-finite. Or when K is algebraically closed and contained in F, in which case G is abelian.