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 chapter on group theory.
When the first edition was written it was fairly uncommon 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 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 conjugacy 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.
|Title: Topics in Algebra|
Author: I. N. Herstein
Publisher: John Wiley & Sons; 2nd edition (June 20, 1975)