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The Cali Garmo does Math


Category: Algebra
Introduction to Group Theory by Ledermann and Weir

Introduction to Group Theory by Ledermann and Weir

Ledermann and Weir take a slightly unique approach in the theory of groups. Their text is slightly difficult to follow in a lot of places as they tend to group things together in non-standard ways. The biggest difference I found with his use of symbols is when talking about homomorphisms. In particular most texts will look at a homomorphism: and when going from to they apply a function such as ; Ledermann and Weird on the other hand choose to use the notation of . Although this format might be more concise, it can at times be a little confusing as to what each symbol is supposed to stand for. My assumption is that he’s trying to show that a homomorphism is similar to group actions with the action being the map , but for an introductory piece, it can be particularly confusing.

The biggest part I did enjoy about this book was the list of topics. Not only did he cover the standard topics of groups, subgroups, cosets, normal groups, conjugacy classes, etc. he also threw in some topics you don’t normally see in an introductory course such as double cosets, free groups, the derived series, soluble and nilpotent groups. He tends to go through topics fairly quickly so it can be a little difficult to follow at times, but it’s not that bad to follow and can generally be followed on your own. Another part about this book I thoroughly enjoyed was that every exercise had a solution in the back of the book. Exercises are essential when learning any new topic and when you are trying to teach yourself, having questions without answers can be a little difficult at times because you can never be sure you’re actually doing it correctly.

All in all I think this book is a really good book for introductory group theory. If you’re willing to invest a little bit of time understanding the notation, you’ll learn more out of this book than most other books on group theory.

Groups and Symmetry by M.A. Armstrong

Groups and Symmetry

Armstrong does an amazing job with his introduction to group theory. Unlike most texts he uses a geometric approach for a lot of his work in groups. With that he uses dihedral groups a lot in his exposition and is always going back to it (and symmetric groups) for examples. I thought it to be refreshing to see group theory presented from an alternative angle. Although he doesn’t provide answers for his exercises, all of the exercises are answerable using the text provided. Some of them were difficult enough to pose some fun challenges while reading the book. Although the book is not the best introductory text I’ve found out there, I thought Armstrong’s introduction of finitely generated abelian groups was very well done. It definitely was better than most of the ones I had seen out there and definitely worth reading more into. The final theorem he put in his book was the Nielsen-Schreier Theorem which states:

Every subgroup of a free group is free

Which is a super fun concept and is not fully introduced in a lot of the texts that I’ve read in the group theory world. If you want a text that does group theory from the slight angle of geometry I definitely recommend this book.