Mark as downloadedDownloaded Algebra Basic Concepts of Abstract Algebra Second edition

Preface to the Second EditionThis second edition has been substantially revised with corrections and improvedpresentations. We have added numerous examples and corrected a few typo errOrS known to the author. We classify finite-generated modules over a principal ideal domain by means of indecomposable modules to simplify the proofs of corrCsponding theorems in the first edition. We update the contents and add new exercises in Sections 2.6 and 2.8. Moreover,Section 5.3 is enriched with some new materialsrelated to homological groups. We now present a detailed descriptions of contents of each chapter. Chapter 1 is devoted to group theory. Aftcr ntroducing thc basic notions of groups and subgroups,we define an action of a group on a set,using which, we prove Sylow’s Thcorems and many other interesting rcsults related to coscts and orbit counting. Some suppermentary exercises are provided to help readers understand the basic theory of linear representations of finite groups. Ther we move on to provethe first(second, third) fundamental theorems of homomorphisms,also called thefirst(second,third) isomorphic theorems. Many conclusions are compared to those from linear algebra. Next we introduce the direct product of groups from the viewpoint of combining some groups into one group,starting with the semidirect product of groups and ending with the Schur-Zassenhaus theorem,which describes a group as a semidirect product of its subgroups via homological methods. We con- clude this chapter with an introduction to simple groups,nilpotent and solvable groups,which will be needed in Chapter 3. In Chapter 2, we introduce the ring theory via representations of rings. Moreprecisely,we focus on modules over a ring and homological properties of rings,such as honological dimensions of rings and modules. After describing the basic prOperties of dimensions of rings and modules,we characterizc rings with(weak) globaldimension zero. Instead of dealing with localizations of a commutative r