Discrete and continuous symmetry via induction and duality

We study the relation of discrete versus continuous symmetry by use of induction/subduction techniques and duality of Abelian groups. We start with translations, then move on to point transformations and to the Euclidean and Non-Euclidean semidirect product groups and their representations. The UIR of discrete space groups are characterized by functions on the dual translation groups which correspond to difference equations on the lattice.