The text begins with a review of group actions and Sylow theory. It includes semidirect products, the Schur-Zassenhaus theorem, the theory of commutators, coprime actions on groups, transfer theory, Frobenius groups, primitive and multiply transitive permutation groups, the simplicity of the PSL groups, the generalized Fitting subgroup and also Thompson's J-subgroup and his normal $p$-complement theorem. Topics that seldom (or never) appear in books are also covered. These include subnormality theory, a group-theoretic proof of Burnside's theorem about groups with order divisible by just two primes, the Wielandt automorphism tower theorem, Yoshida's transfer theorem, the ``principal ideal theorem'' of transfer theory and many smaller results that are not very well known. Proofs often contain original ideas, and they are given in complete detail. In many cases they are simpler than can be found elsewhere. The book is largely based on the author's lectures, and consequently, the style is friendly and somewhat informal. Finally, the book includes a large collection of problems at disparate levels of difficulty. 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. He received the University of Wisconsin Distinguished Teaching Award in 1985, the Benjamin Smith Reynolds Teaching Award in 1989, and the Wisconsin Section MAA Teaching Award in 1993, to name only a few. He was also honored by being the selected MAA Polya Lecturer in 2003-2005.
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"As might be expected from a book by Isaacs, although the topics are familiar, almost every chapter contains unexpected gems not to be found in most basic textbooks. Surely many readers will be inspired by this book to continue their study of the fascinating field of finite group theory." ---- Mathematical Reviews
"This is one serious group theory book, intended for graduate students with strong algebra backgrounds who plan to read papers on group theory after this course. Each chapter comes with a freight car full of substantial exercises, ranging in difficulty from trivial to research level. If you're studying group theory anytime soon get a copy." ---- MAA Reviews
"This is one serious group theory book, intended for graduate students with strong algebra backgrounds who plan to read papers on group theory after this course. Each chapter comes with a freight car full of substantial exercises, ranging in difficulty from trivial to research level. If you're studying group theory anytime soon get a copy." ---- MAA Reviews
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Excellent intermediate text on finite groups excluding representation theory. The proofs are succinct and insightful but better still, the motivation for the development of the subject is clear and engaging. Covers similar material to The Theory of Finite Groups: An Introduction (Universitext) by Hans Kurzweil and Bernd Stellmacher but it is well worth having both.
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Inside This Book
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Key Phrases - Statistically Improbable Phrases (SIPs):
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zipper lemma, unique smallest normal subgroup, action via automorphisms, focal subgroup theorem, nilpotence class, automorphism tower, conjugacy part, coprime actions, unique maximal subgroup, orbital graph, primitive permutation group, minimal normal subgroup, right transversal, corresponding factor group, subnormal subgroups, unique normal subgroup, full stabilizer, transfer homomorphism, point stabilizer, upper central series, nonabelian simple group, nonidentity elements, strong conjugates, permutation isomorphic, elementary abelian subgroup Key Phrases - Capitalized Phrases (CAPs): (learn more)
Proof of Theorem, A-invariant Sylow, Sylow C-theorem, Proof of Lemma, Use Problem, Hall-Higman Lemma, Sylow E-theorem Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
zipper lemma, unique smallest normal subgroup, action via automorphisms, focal subgroup theorem, nilpotence class, automorphism tower, conjugacy part, coprime actions, unique maximal subgroup, orbital graph, primitive permutation group, minimal normal subgroup, right transversal, corresponding factor group, subnormal subgroups, unique normal subgroup, full stabilizer, transfer homomorphism, point stabilizer, upper central series, nonabelian simple group, nonidentity elements, strong conjugates, permutation isomorphic, elementary abelian subgroup Key Phrases - Capitalized Phrases (CAPs): (learn more)
Proof of Theorem, A-invariant Sylow, Sylow C-theorem, Proof of Lemma, Use Problem, Hall-Higman Lemma, Sylow E-theorem Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
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