It seems appropriate to begin this book with a topic that underlies virtually
all of finite group theory: the Sylow theorems. In this chapter, we state and
prove these theorems, and we present some applications and related results.
Although much of this material should be very familiar, we suspect that
most readers will find that at least some of the content of this chapter is
new to them.
Although the theorem that proves Sylow subgroups always exist dates
back to 1872, the existence proof that we have decided to present is that
of H. Wielandt, published in 1959. Wielandt's proof is slick and short, but
it does have some drawbacks. It is based on a trick that seems to have
no other application, and the proof is not really constructive; it gives no
guidance about how, in practice, one might actually find a Sylow subgroup.
But Wielandt's proof is beautiful, and that is the principal motivation for
presenting it here.
Also, Wielandt's proof gives us an excuse to present a quick review of the
theory of group actions, which are nearly as ubiquitous in the study of finite
groups as are the Sylow theorems themselves. We devote the rest of this
section to the relevant definitions and basic facts about actions, although
we omit some details from the proofs.
Let G be a group, and let fibea nonempty set. (We will often refer to
the elements of ft as "points".) Suppose we have a rule that determines a
new element of ft, denoted a«g, whenever we are given a point a G ft and
an element g G G. We say that this rule defines an action of G on ft if the
following two conditions hold.