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
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
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