Decoupled motions in parallel manipulators lead to simpler mechanical structure, larger workspace, lower cost, higher accuracy and easier control as well as designs with lower mobility. One of these manipulators is 3T1R parallel manipulator with general translation in space and a rotation about a fixed direction. Designers used systematic methods to synthesis decoupled parallel manipulators. Screw theory with wide applications has been used to the type synthesis of the parallel manipulators. Moreover, synthesis of uncoupled parallel manipulators has widely studied, while the synthesis of decoupled parallel manipulators has received less attention. One type of decoupled parallel manipulators is a manipulator with group-uncoupling motions whose outputs can be controlled by a series of actuated joints. Here, using screw theory, group-uncoupling motion of 3T1R non-redundant degree of freedoms of the parallel manipulator with the first actuated joints within every four legs is introduced. Upon comparison of the kinematic equations governing the several groups of 3T1R uncoupled parallel manipulators and their Jacobian matrices derived based on screw theory. The geometric conditions of the actuated wrenches are obtained as well as the comprehensive definitions of group-uncoupling motions. Finally, these results have been clarified the related feasible legs of each groups to fulfill the required motions.