An Introduction to the Theory of Groups (Graduate Texts in Mathematics (148))

My second abstract algebra class had no lecture notes, and the textbook was Robinson's A Course in the Theory of Groups. I couldn't get through even the first chapter of this book, so my professor recommended that I read this book by Rotman instead. The structure of things in algebra started making much more sense to me. I was able to understand much better what was going on in class and the motivation for what we studied. Though it's self-contained, I'd recommend this as a book for self-study once you've already got a handle on the basics of algebra. The exercises are tricky but doable, and the way the book is structured, it is essential to read them to understand the arguments. Some proofs depend on solutions to exercises. Anyone Who Has Studied Abstract Algebra And Linear Algebra As An Undergraduate Can Understand This Book. This Edition Has Been Completely Revised And Reorganized, Without However Losing Any Of The Clarity Of Presentation That Was The Hallmark Of The Previous Editions. The First Six Chapters Provide Ample Material For A First Course: Beginning With The Basic Properties Of Groups And Homomorphisms, Topics Covered Include Lagrange's Theorem, The Noether Isomorphism Theorems, Symmetric Groups, G-sets, The Sylow Theorems, Finite Abelian Groups, The Krull-schmidt Theorem, Solvable And Nilpotent Groups, And The Jordan-holder Theorem. The Middle Portion Of The Book Uses The Jordan-holder Theorem To Organize The Discussion Of Extensions (automorphism Groups, Semidirect Products, The Schur-zassenhaus Lemma, Schur Multipliers) And Simple Groups (simplicity Of Projective Unimodular Groups And, After A Return To G-sets, A Construction Of The Sporadic Mathieu Groups). Groups And Homomorphisms -- The Isomorphism Theorems -- Symmetric Groups And G-sets -- The Sylow Theorems -- Normal Series -- Finite Direct Products -- Extensions And Cohomology -- Some Simple Linear Groups -- Permutations And Mathieu Groups -- Abelian Groups -- Free Groups And Free Products -- The Word Problem -- Appendices: Some Major Algebraic Systems -- Equivalence Relations And Equivalence Classes -- Functions -- Zorn's Lemma -- Countability -- Commutative Rings -- Bibliography -- Notation -- Index. By Joseph J. Rotman.

Anyone who has studied abstract algebra and linear algebra as an undergraduate can understand this book. The first six chapters provide material for a first course, while the rest of the book covers more advanced topics. This revised edition retains the clarity of presentation that was the hallmark of the previous editions.

From the reviews:

"Rotman has given us a very readable and valuable text, and has shown us many beautiful vistas along his chosen route." —MATHEMATICAL REVIEWS

Groups And Homomorphisms -- The Isomorphism Theorems -- Symmetric Groups And G-sets -- The Sylow Theorems -- Normal Series -- Finite Direct Products -- Extensions And Cohomology -- Some Simple Linear Groups -- Permutations And The Mathieu Groups -- Abelian Groups -- Free Groups And Free Products -- The Word Problem. Joseph Rotman. Includes Bibliographical References (p. [495]-497) And Indexes. Generalizations of the quadratic formula for cubic and quartic polynomials were discovered in the sixteenth century, and one of the major mathematical problems thereafter was to find analogous formulas for the roots of polynomials of higher degree; all attempts failed.