p-Groups and the Sylow Theorems
Mathematics is the tool specially suited for dealing with abstract concepts of any kind. There is no limit to its power in this field.
—Paul Adrien Maurice Dirac
Historically, the theory of groups was concerned only with groups of permutations of a set. This point of view is reinforced by Cayley's theorem, which shows that every abstract group can be viewed as a subgroup of a group of permutations. The concept of an abstract group became important because it focuses attention on those aspects of a group of permutations that do not depend on the underlying set. However, this abstract formulation of the theory loses sight of the combinatorial aspects that are more in evidence for groups of permutations. And these counting methods give important information about abstract groups. The best example is Lagrange's theorem, which is based on the fact that a subgroup partitions the group into cosets each having the same number of elements as the subgroup.
In Section 8.2, we derive another such counting theorem, the class equation, from a partition of a finite group and use it, among other things, to deduce many properties of groups of prime power order. Then, in Section 8.3, we present a far-reaching counting method that includes the proof of Lagrange's theorem and the class equation and which, in Section 8.4, we use to prove the Sylow theorems. These beautiful results guarantee the presence of subgroups of prime power order in every finite group ...