BEST TOP USEFUL BOOK FOR THE FOLLOWING EXAMS AND INTERVIEWS AND RESEARCH* ALL INTERNATIONAL EDUCATION RESEARCH AND EXAMS* ALL UG & PG & PHD & PDF * IIT JEE* IIT JAM* GATE* CIVILS SERVICES* SET - STATE ELIGIBILITY TEST* NET - NATIONAL ELIGIBILITY TEST* ALL INDIAN UNIVERSITY EXAMS* CENTRAL UNIVERSITY EXAMS* IISC ENTRANCE EXAMS* INSPIRE FELLOWSHIP* ALL FELLOWSHIP NATIONAL LEVEL EXAMS* POST DOCTORAL EXAMS* ALL INDIAN UNIVERSITY ENTRANCE EXAMS* ASSISTANT PROFESSOR EXAMS* JUNIOR LECTURER EXAMS* RESEARCH ORIENTED* SUBJECT ANALYSIS* SUBJECT THESISPreface to the Second Edition I approached rev1smg Topics in Algebra with a certain amount of trepidation. On the whole, I was satisfied with the first edition and did not want to tamper with it. However, there were certain changes I felt should be made, changes which would not affect the general style or content, but which would make the book a little more complete. I hope that I have achieved this objective in the present version. For the most part, the major changes take place in the chapt¥r on group theory. When the first edition was written it was fairly un-common for a student learning abstract algebra to have had any previous exposure to linear algebra. Nowadays quite the opposite is true; many students, perhaps even a majority, have learned something about 2 x 2 matrices at this stage. Thus I felt free here to draw on 2 x 2 matrices for examples and problems. These parts, which depend on some knowledge of linear algebra, are indicated with a #. In the chapter on groups I have largely expanded one section, that on Sylow's theorem, and added two others, one on direct products and one on the structure of finite abelian groups. In the previous treatment of Sylow's theorem, only the existence of a Sylow subgroup was shown. This was done following the proof of Wielandt. The conjugacy of the Sylow subgroups and their number were developed in a series of exercises, but not in the text proper. Now all the parts of Sylow's theorem are done in the text materi9-l. In addition to the proof previously given for the existence, two other proofs of existence are carried out. One could accuse me of overkill at this point, probably rightfully so. The fact of the matter is that Sylow's theorem is important, that each proof illustrates a different aspect of group theory and, above all, that I love Sylow's theorem. The proof of the con-jugacy and number of Sylow subgroups exploits double cosets. A by-product of this development is that a means is given for finding Sylow subgroups in a large set of symmetric groups. For some mysterious reason known only to myself, I had omitted direct products in the first edition. Why is beyond me. The material is easy, straightforward, and important. This lacuna is now filled in the section treating direct products. With this in hand, I go on in the next section to prove the decomposition of a finite abelian group as a direct product of cyclic groups and also prove the uniqueness of the invariants associated with this decomposition. In point of fact, this decomposition was already in the first edition, at the end of the chapter on vector spaces, as a consequence of the structure of finitely generated modules over Euclidean rings. However, the case of a finite group is of great importance by itself; the section on finite abelian groups underlines this importance. Its presence in the chapter on groups, an early chapter, makes it more likely that it will be taught. One other entire section has been added at the end of the chapter on field theory.