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Commutators C h a p t e r 5. Permutation Groups C h a p t e r 9. M o r e Transfer T h e o r y Appendix: Index T h e Basics Preface T h i s b o o k is a somewhat expanded version of a graduate course i n finite group theory that I often teach at the U n i v e r s i t y of W i s c o n s i n. I offer this course i n order to share what I consider to be a beautiful subject w i t h as m a n y people as possible, and also to provide the solid b a c k g r o u n d i n pure group theory t h a t m y d o c t o r a l students need to carry out their thesis work i n representation theory.
T h e focus of group theory research has changed profoundly i n recent decades. S t a r t i n g near the beginning of the 20th century w i t h the work of W.
B u r n s i d e , the major p r o b l e m was to find a n d classify the finite simple groups, and indeed, m a n y of the most significant results i n pure group theory and i n representation theory were directly, or at least peripherally, related to this goal. T h e simple-group classification now appears to be complete, and current research has shifted to other aspects of finite group theory i n c l u d i n g p e r m u t a t i o n groups, p-groups and especially, representation theory.
It is certainly no less essential i n this post-classification p e r i o d that group-theory researchers, whatever their subspecialty, s h o u l d have a mastery of the classical techniques and results, and so w i t h o u t a t t e m p t i n g to be encyclopedic, I have included m u c h of t h a t m a t e r i a l here.
B u t m y choice of topics was largely determined by m y p r i m a r y goal i n w r i t i n g this book, w h i c h was to convey to readers m y feeling for the beauty and elegance of finite group theory. G i v e n its origin, this b o o k should certainly be suitable as a text for a graduate course like mine.
B u t I have t r i e d to write it so that readers w o u l d also be comfortable using it for independent study, a n d for t h a t reason, I have t r i e d to preserve some of the informal flavor of m y classroom. I have tried to keep the proofs as short and clean as possible, b u t w i t h o u t o m i t t i n g ix X Preface details, and indeed, i n some of the more difficult m a t e r i a l , m y arguments are simpler t h a n can be found i n p r i n t elsewhere.
F i n a l l y , since I firmly believe t h a t one cannot learn mathematics w i t h o u t doing it, I have included a large number of problems, m a n y of w h i c h are far from routine. Some of the m a t e r i a l here has rarely, if ever, appeared previously i n books. Just i n the first few chapters, for example, we offer Zenkov's marvelous theorem about intersections of abelian subgroups, W i e l a n d t ' s "zipper l e m m a " i n s u b n o r m a l i t y theory and a proof of Horosevskii's theorem that the order of a group a u t o m o r p h i s m can never exceed the order of the group.
L a t e r chapters include m a n y more advanced topics that are h a r d or impossible to find elsewhere. M o s t of the students who attend m y group-theory course are second-year graduate students, w i t h a substantial m i n o r i t y of first-year students, and an occasional well-prepared undergraduate. A l m o s t 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 b o o k w i l l have a similar background, and so I have decided not to begin at the beginning. M o s t of m y readers like m y students w i l l have previously seen basic group theory, so I wanted to avoid repeating that m a t e r i a l and to start w i t h something more exciting: Sylow theory. B u t I recognize that m y audience is not homogeneous, and some readers w i l l have gaps i n their preparation, so I have included an a p p e n d i x that contains most of the assumed m a t e r i a l in a fairly condensed form.
O n the other hand, I expect that m a n y i n m y audience w i l l already know the Sylow theorems, but I a m confident that even these well-prepared readers w i l l find m a t e r i a l that is new to t h e m w i t h i n the first few sections. M y semester-long graduate course at W i s c o n s i n covers most of the first seven chapters of this book, starting w i t h the Sylow theorems and culm i n a t i n g w i t h a purely group-theoretic proof of B u r n s i d e ' s famous p q theorem.
Some of the topics along the way are s u b n o r m a l i t y theory, the Schur-Zassenhaus theorem, transfer theory, coprime group actions, Frobenius groups, and the n o r m a l p-complement theorems of Frobenius and of T h o m p s o n.
T h e last three chapters cover material for w h i c h I never have time i n class. F i n a l l y , C h a p t e r 10 presents some advanced topics i n transfer theory, i n c l u d i n g Y o s h i d a ' s theorem a n d the so-called "principal ideal theorem". F i n a l l y , I t h a n k m y m a n y students and colleagues who have contributed ideas, suggestions and corrections while this b o o k was being w r i t t e n.
Chapter 1 Sylow Theory 1A It seems appropriate to begin this b o o k w i t h a t o p i c t h a t underlies v i r t u a l l y all of finite group theory: the Sylow theorems. I n this chapter, we state and prove these theorems, a n d we present some applications a n d related results. A l t h o u g h the theorem t h a t proves Sylow subgroups always exist dates back to , the existence proof t h a t we have decided to present is t h a t of H.
W i e l a n d t ' s proof is slick a n d short, but it does have some drawbacks. It is based o n a t r i c k t h a t seems to have no other a p p l i c a t i o n , a n d the proof is not really constructive; it gives no guidance about how, i n practice, one might a c t u a l l y find a Sylow subgroup. A l s o , W i e l a n d t ' s proof gives us an excuse to present a quick review of the theory of group actions, w h i c h are nearly as u b i q u i t o u s i n the study of finite groups as are the Sylow theorems themselves.
W e devote the rest of this section to the relevant definitions a n d basic facts about actions, a l t h o u g h we o m i t some details from the proofs. We w i l l often refer to the elements of ft as "points".
Suppose we have a rule t h a t determines a new element of ft, denoted a-g, whenever we are given a point a e ft and a n element g e G. W e say t h a t this rule defines a n a c t i o n of G on ft i f the following two conditions hold. Suppose t h a t G acts on ft. See the problems at the end of this section.
Before we proceed i n that direction, however, it seems appropriate to mention a few examples. It follows t h a t the corresponding p e r m u t a t i o n representation of G is an i s o m o r p h i s m of G into S y m G , and this proves C a y l e y ' s theorem: every group is isomorphic to a group of permutations on some set.
T h e kernel, therefore, is the center Z G. O f course, i n order to make these examples work, we do not really need ft to be a l l subsets of G.
F o r example, since a conjugate of a subgroup is always a subgroup, the conjugation action is well defined if we take ft to be the set of a l l subgroups of G. A l s o , b o t h right m u l t i p l i c a t i o n 9 9 1A 3 a n d conjugation preserve cardinality, and so each of these actions makes sense i f we take ft to be the collection of a l l subsets of G of some fixed size.
In fact, as we shall see, the t r i c k i n W i e l a n d t ' s proof of the Sylow existence theorem is to use the right m u l t i p l i c a t i o n action of G on its set of subsets w i t h a certain fixed cardinality.
We mention one other example, w h i c h is a special case of the rightm u l t i p l i c a t i o n action on subsets that we discussed i n the previous paragraph. If X is any right coset of H , it is easy to see that X g is also a right coset of H. It is easy to check t h a t G is a subgroup of G ; it is called the stabilizer of the point a.
For example, i n the regular action of G on itself, the stabilizer of every point element of G is the t r i v i a l subgroup. In the conjugation action of G on G, the stabilizer of x G G is the centralizer C x and i n the conjugation a c t i o n of G on subsets, the stabilizer of a subset X is the normalizer N X.
A useful general fact about point stabilizers is the following, w h i c h is easy to prove. T h i s is because two cosets H u and H v are identical if and only if u G H v. It follows t h a t the kernel of the action of G on the right cosets of H i n l l x x G is exactly f H x. T h i s subgroup is called the core of H i n G, denoted xeG c o r e H.
T h e core of H is n o r m a l i n G because i t is the kernel of an action, and, clearly, it is contained i n H. In fact, if AT x x G We have digressed from our goal, w h i c h is to actions to count things. B u t having come this far, results that our discussion has essentially proved.
U In order to pursue our m a i n goal, w h i c h is counting, we need to discuss the "orbits" of an action. A l s o , since every point is i n at least one orbit, it follows t h a t the orbits of the action of G on ft p a r t i t i o n ft.
We mention some examples of orbits a n d orbit decompositions. It is easy to see t h a t the orbits of this action are exactly the left cosets of H i n G. We leave to the reader the p r o b l e m of realizing the right cosets of H in G as the orbits of an appropriate action of H. B u t be careful: the rule x - h - hx does n o t define an action.
Perhaps it is more interesting to consider the conjugation action of G on itself, where the orbits are exactly the conjugacy classes of G. How 1. T h e key result here is the following. The surjectivity is easy, a n d we do t h a t first. T h i s proves t h a t 6 is injective, as required. M o r e informally, this says t h a t the actions of G on A and on O are "essentially the same". Since every action can be thought of as composed of the actions on the i n d i v i d u a l orbits, and each of these actions is p e r m u t a t i o n isomorphic to the rightm u l t i p l i c a t i o n action of G on the right cosets of some subgroup, we see t h a t these actions on cosets are t r u l y fundamental: every group action can be viewed as being composed of actions on right cosets of various subgroups.
We close this section w i t h two familiar a n d useful applications of the fundamental counting principle. T h e conjugates of H form a n orbit under the conjugation action of G on the set of subsets of G. Observe t h a t H g K is a u n i o n of right cosets of H , and t h a t these cosets form an orbit under the action of K. Proofs of these useful facts appear i n the appendix, but we suggest t h a t readers t r y to find their o w n arguments.
A l s o , recall t h a t the product H K of subgroups H and K is not always a subgroup. This too is proved i n the appendix. A n action of a group G on a set ft is transitive i f ft consists of a single orbit. T h e nonnegative-integer-valued function is called the p e r m u t a t i o n character associated w i t h the action.
T h u s the number of orbits is 1 1 geG w h i c h is the average value of x over the group. L e t G be a finite group, and suppose that H 2 t f , where runs over h e H. Use this information to get a n estimate o n elements of G where vanishes.
L e t ft be the set of n-tuples [x ,x , Show that divides the number of C - o r b i t s of size 1 on ft, and deduce t h a t the number of elements of order p i n G is congruent to - 1 mod p.
C a u c h y ' s theorem can also be derived as a corollary of Sylow's theorem. Show t h a t G has a unique subgroup of order p. G IB F i x a prime number p. A finite group whose order is a power of p is called a pgroup. It is often convenient, however, to use this nomenclature somewhat carelessly, and to refer to a group as a "p-group" even if there is no p a r t i c u l a r prime p under consideration.
F o r example, i n proving some theorem, one might say: it suffices to check that the result holds for p-groups. W h a t is meant here, of course, is that it suffices to show t h a t the theorem holds for all p-groups for a l l primes p. We m e n t i o n that, a l t h o u g h i n this book a p-group is required to be finite, it is also possible to define infinite p-groups.
T h e more general definition is t h a t a not necessarily finite group G is a p-group if every element of G has finite p-power order.
Otherwise the book is entirely self-contained. The book follows both strands of the theory: the exceptional characteristics of Suzuki and Feit and the block character theory of Brauer and includes refinements of original proofs that have become available as the subject has grown. By the late s, considerable advances to use Burnside's words had been made in pure finite group theory. In particular, the Thompson subgroup, and Thompson's techniques of "local analysis" had been developed, and these had proved Topics include representation theory of rings with identity, representation theory of finite groups, applications of the theory of characters, construction of irreducible representations and modular representations.
Skip to content. Topics that seldom or never appear in books are also covered. Proofs often contain original ideas, and they are given in complete detail. In many cases they are simpler than can be found elsewhere. These should enable students to practice group theory and not just read about it.
Martin Isaacs is professor of mathematics at the University of Wisconsin, Madison. Over the years, he has received many teaching awards and is well known for his inspiring teaching and lecturing. Even though representation theory and constructions of simple groups have been omitted, the text serves as a springboard for deeper study in many directions.
Its proofs often have elegance and crystalline beauty. This textbook is intended for the reader who has been exposed to about three years of serious mathematics. The notion of a group appears widely in mathematics and even further afield in physics and chemistry, and the fundamental idea should be known to all mathematicians. In this textbook a purely algebraic approach is taken and the choice of material is based upon the notion of conjugacy.
The aim is not only to cover basic material, but also to present group theory as a living, vibrant and growing discipline, by including references and discussion of some work up to the present day.
In particular, the theory has been a key ingredient in the classification of finite simple groups. Characters are also of interest in their own right, and their properties are closely related to properties of the structure of the underlying group. The book begins by developing the module theory of complex group algebras. After the module-theoretic foundations are laid in the first chapter, the focus is primarily on characters. This enhances the accessibility of the material for students, which was a major consideration in the writing.
Also with students in mind, a large number of problems are included, many of them quite challenging. In addition to the development of the basic theory using a cleaner notation than previously , a number of more specialized topics are covered with accessible presentations.
These include projective representations, the basics of the Schur index, irreducible character degrees and group structure, complex linear groups, exceptional characters, and a fairly extensive introduction to blocks and Brauer characters. It is largely self-contained, requiring of the reader only the most basic facts of linear algebra, group theory, Galois theory and ring and module theory.
ISBN: Category: Mathematics Page: View: The theory of finite simple groups enjoyed a period of spectacular activity in the s and s. The first edition of Gorenstein's book was published in , at the time of some of the first major classification results. The second edition was published in , when it was clear that the classification was understood and the proof was within reach.
Gorenstein's treatment of the subject proved prescient, as many of the developments between the two editions could be seen as continuations of the material in the book. It reports on the great progress achieved since through the joint effort of researchers in both areas. The book allows access to the results achieved so far and aims to increase the scientific exchange between number theory and group theory. The present volume and its successor have therefore the more modest aim of giving descriptions of the recent development of certain important parts of the subject, and even in these parts no attempt at completeness has been made.
Chapter VII deals with the representation theory of finite groups in arbitrary fields with particular attention to those of non-zero charac teristic. That part of modular representation theory which is essentially the block theory of complex characters has not been included, as there are already monographs on this subject and others will shortly appear.
Instead, we have restricted ourselves to such results as can be obtained by purely module-theoretical means. This is achieved by mainly keeping the required background to the level of undergraduate linear algebra, group theory and very basic ring theory. Module theory and Wedderburn theory, as well as tensor products, are deliberately avoided.
Instead, we take an approach based on discrete Fourier Analysis. Applications to the spectral theory of graphs are given to help the student appreciate the usefulness of the subject. A number of exercises are included. However, it can also be used as a reference for workers in all areas of mathematics and statistics.
He has also had a great influence on the development of algebra, and particularly group theory in China. The papers contain the main general results, as well as recent ones, on certain topics within this discipline.
The chief editor, Zhe-Xian Wan, is a leading algebraist in China.
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