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Preface

details, and indeed, in some of the more difficult material, my arguments are

simpler than can be found in print elsewhere. Finally, since I firmly believe

that one cannot learn mathematics without doing it, I have included a large

number of problems, many of which are far from routine.

Some of the material here has rarely, if ever, appeared previously in

books. Just in the first few chapters, for example, we offer Zenkov's mar-

velous theorem about intersections of abelian subgroups, Wielandt's "zipper

lemma" in subnormality theory and a proof of Horosevskii's theorem that

the order of a group automorphism can never exceed the order of the group.

Later chapters include many more advanced topics that are hard or impos-

sible to find elsewhere.

Most of the students who attend my group-theory course are second-year

graduate students, with a substantial minority of first-year students, and an

occasional well-prepared undergraduate. Almost all of these people had

previously been exposed to a standard first-year graduate abstract algebra

course covering the basics of groups, rings and fields. I expect that most

readers of this book will have a similar background, and so I have decided

not to begin at the beginning.

Most of my readers (like my students) will have previously seen basic

group theory, so I wanted to avoid repeating that material and to start with

something more exciting: Sylow theory. But I recognize that my audience

is not homogeneous, and some readers will have gaps in their preparation,

so I have included an appendix that contains most of the assumed material

in a fairly condensed form. On the other hand, I expect that many in my

audience will already know the Sylow theorems, but I am confident that even

these well-prepared readers will find material that is new to them within the

first few sections.

My semester-long graduate course at Wisconsin covers most of the first

seven chapters of this book, starting with the Sylow theorems and cul-

minating with a purely group-theoretic proof of Burnside's famous

paqb-

theorem. Some of the topics along the way are subnormality theory, the

Schur-Zassenhaus theorem, transfer theory, coprime group actions, Frobe-

nius groups, and the normal p-complement theorems of Frobenius and of

Thompson. The last three chapters cover material for which I never have

time in class. Chapter 8 includes a proof of the simplicity of the groups

PSL(n, q), and also some graph-theoretic techniques for studying subdegree&

of primitive and nonprimitive permutation groups. Subnormality theory is

revisited in Chapter 9, which includes Wielandt's beautiful automorphism

tower theorem and the Thompson-Wielandt theorem related to the Sims