The manipulation of rigid bodies by manipulators which are motor driven kinematic chains is a fundamental aspect of robotics. In this paper, we discuss the kinematics of such processes and discuss the classification of kinematic chains using ideas from algebra and group theory. Earlier work on the role of Lie groups in mechanisms is contained in Herve [7], but the role of Lie algebras is not considered by this author. More relevant (but less group theoretic) is the extensive case-by-case analysis found in Pieper's thesis [9]. In fact, Pieper's work suggests an interesting and rather general problem in Galois theory which is directly related to manipulation. Also of interest is the well-known Baker-Campbell-Hausdorff formula for the derivative of a product of exponentials since such products are of fundamental importance in the study of kinematic programming.