ABSTRACT
17.1 In many theories one tries to understand the global phenomena by looking at a kind of “atoms”, i.e. smaller points of the puzzle that can be glued together to catch the big picture. In the theory of numbers the prime numbers may be viewed as such atoms of the theory, for polynomials the irreducible polynomials may serve the same aim, for modules over rings the irreducible modules or simple modules turn up leading to irreducible representations, playing the same role in representation theory of groups (see 34.13, 34.14 and Theorem 34.15). For group theory the building blocks we look for are p-groups and the Sylow theorems tell us how the structure of the group is partially reflected in the presence of certain specific p-subgroups i.e. Sylow subgroups. The information encoded in Sylow subgroups is less precise than what happens with the prime decomposition of a number, in particular a group need not be the product of its Sylow subgroups (this only happens for nilpotent groups) but nevertheless the study of Sylow subgroups is a powerful tool in general group theory for finite subgroups. For a full classification of even simple finite groups a lot more methods are needed; the fact that simple finite groups have been classified is one of the big achievements in Mathematics of the foregoing century.
