But before Carter gets to any of that, he has five full chapters dedicated to defining, motivating, and giving examples of groups, and this is where the difference in his approach shows through most clearly. In the preface, Carter emphasizes that he calls it a book “ about group theory because this book does not aim to cover group theory comprehensively.” Given that disclaimer, I was surprised to find that most of the topics I would cover in a first semester of group theory are indeed covered in this book: subgroups, Lagrange’s Theorem, products and quotients, conjugacy, homomorphisms, the Fundamental Theorem of Abelian Groups, Sylow’s Theorems, and Galois theory. To be precise, Carter’s book is not a full abstract algebra book but rather a book about group theory with a few topics about fields covered at the end. However, Nathan Carter’s Visual Group Theory, published this summer by the MAA, has done just that. It therefore seems like it would be hard for a new book to come out with a new viewpoint on the field. In any event, there are many books on the market that one could choose from, with a large amount of variation in the topics they cover and the approaches they use.
Maybe they prefer the more traditional groups-first approach or maybe they prefer the rings-first approach. Most of the mathematicians I know already have an opinion on the question “What is the best book from which to learn abstract algebra?” Maybe it is the book they used as undergraduates, or maybe it is a book they have taught from.
0 kommentar(er)
