The Sylow theorems show that the groups of prime power order play a distinguished role in the theory of finite groups. Admittedly, the construction of a finite group from its Sylow subgroups is generally a very difficult task. In the case of solvable groups, the theory of Sylow systems developed by P. Hall and the theory of p-length developed by P. Hall and G. Higman give significant insight; we will discuss these theories in Chapter VI. For simple groups, any understanding of their construction from Sylow subgroups is still in its infancy. However, the work of Thompson on simple groups has given significant insight and guided the theory of p-groups in new directions.