Chapter 1

Sylow Theory

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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.

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http://dx.doi.org/10.1090/gsm/092/01