 tasks for which the existence of a wait-free read-write protocol is undecidable. Here we give a complete classification of loop agreement tasks. Each loop agreement task can be assigned an _algebraic signature_ consisting of a group   and a distinguished element   in  . Remarkably, this signature completely characterizes the task's computational power. If   and   are loop agreement tasks with respective signatures   and  , then   implements   if and only if there exists a group homomorphism   carrying   to  . In short, the algorithmic problem of determining how to implement one loop agreement task in terms of another reduces to a problem in group theory.  