Math 505: Algebra I. Generalizations of symmetry, Group Theory.

Math 506: Algebra II. Number systems and their generalizations: Rings and Fields. Applications to
Information and Coding Theory.

1. (505) Geoff C. Smith, Olga M. Tabachnikova, Topics in Group Theory, (Springer Undergraduate Mathematics Series), $34.95, Paperback - 256 pages (August 2000), Springer Verlag; ISBN: 1852332352. (ZBL Review: This book, an undergraduate text on group theory, is not aimed at the complete beginner; it was designed to support students already engaged in a first group theory course.
        The basics of the theory are covered in detail (finite groups, including actions and the Sylow theorems, the structure of finitely generated Abelian groups, various series with applications and the Jordan-Hoelder theorem).
       The usual section devoted to applications of the theory to groups of rather small order is there, too.
      What is new, however, in this type of text, is the extent given to illustrating the theory through considering infinite groups. The last part of the book, about seventy pages long, is dealing with various classes of infinite nonabelian groups, presentations and free groups.
      The authors succeed to simplify a number of standard proofs through fresh elementary arguments. Very lucid writing provides the reader with reasons for performing every step in the proofs.
      But what sets this book apart from the crowd is its sparkling style. To some, the style could appear as irreverent and a bit intrusive. To this reviewer, reading the book was a delight, lessened only by the envy of not having written it himself.
     The book is not free from misprints and small glitches. The process of detecting them is still in progress and a web address containing their list is provided in the preface. They will be for sure corrected in the forthcoming editions.
     To summarize: this is one of the best texts in mathematics I got my hands on.
[ M.Deaconescu (Safat) ]

2. (505-506) Joseph A. Gallian, Contemporary Abstract Algebra, $84.75, Hardcover 4th edition (June 1998), Houghton Mifflin College; ISBN: 0395861799. (Used by G. Liu.)

3. (505-506) Israel N. Herstein, Topics in Algebra, $96.95, Hardcover 2nd edition (June 1975), John Wiley & Sons; ISBN: 0471010901.

4. (505-506) D. A. R. Wallace, Groups, Rings and Fields, (Springer Undegraduate Mathematics Series), $34.95, Paperback, 237 pages (September 1998), Springer Verlag; ISBN: 3540761772 (Customer reviews at amazon.com are not very good.)

5. (506) B. A. Sethuraman, Rings, Fields, and Vector Spaces : An Introduction to Abstract Algebra Via Geometric Constructibility (Undergraduate Texts in Mathematics), $34.95, Hardcover, 160 pages (January 1997), Springer Verlag; ISBN: 0387948481. (Book Description: Using the proof of the non-trisectability of an arbitrary angle as a final goal, the author develops in an easy conversational style the basics of rings, fields, and vector spaces. Originally developed as a text for an introduction to    algebra course for future high-school teachers at California State University, Northridge, the focus of this book is on exposition. It would serve extremely well as a focused, one-semester introduction to abstract algebra.)

6. (505-506) Victor P. Snaith, Groups, Rings and Galois Theory, $32.00, Hardcover (November 1998), 172 pages, World Scientific Pub Co; ISBN: 9810235089.

7. (506) John A. Beachy, Introductory Lectures on Rings and Modules (London Mathematical Society Student Texts, 47), $25.95, Paperback (August 1999), Cambridge Univ Pr (Pap Txt); ISBN: 0521644070.(ZBL review: This introduction to rings and modules contains, apart from the basic notions, a chapter on representations of finite groups and one on non-commutative rings, going as far as orders in simple Artinian rings. There is no mention of exact sequences or even of categories; this makes some definitions such as tensor product rather laboured and the situation is not helped by an absence of motivation. Rings are defined with unit element, but their direct product is called direct sum, which leads to difficulties in considering subrings, difficulties the author avoids by not going into detail. The basic correspondence between vector space endomorphisms and matrices is relegated to an aside, whereas the proof of associativity of matrix multiplication is given twice (word for word the same). As a result the book is not easy to read but one gets the impression that a small amount of editing could bring a great improvement.
[ P.M.Cohn (London) ]

8. (505-506) J. BEACHY & W. BLAIR, ABSTRACT ALGEBRA, $44.95, WAVELAND PRESS PUBL., CO., 2ND ED., 1995, ISBN: 0881338664 .

9. W. E. Deskins, Abstract Algebra (Dover Books on Mathematics), $12.76, Paperback Rpt uab edition (February 1996), Dover Pubns; ISBN: 0486688887.

10. T.A. Whitelaw, Introduction to Abstract Algebra, CRC Press, Third Edition, 1995, $44.95,
256 pp.