Learn Group Theory: Sources & Resources

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Discussion Overview

The discussion centers around resources for learning group theory, particularly in relation to its applications in physics and mathematics. Participants seek recommendations for books and other materials that cater to different learning goals and levels of understanding.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants suggest that basic courses in abstract algebra may be beneficial for understanding group theory.
  • Concerns are raised about the sufficiency of short notes on group theory for deep understanding.
  • Multiple participants emphasize the importance of personal learning goals, such as whether one seeks to understand the mathematics rigorously or is more interested in applications like representation theory in quantum field theory (QFT).
  • Questions are posed regarding the participant's background in set theory and proofs, as well as their interest in discrete versus continuous groups.
  • Recommendations for specific books include Brian Hall's book on Lie groups, Georgi's work for physical applications, and Zee's recent publication, though familiarity with Zee's book is limited among participants.
  • One participant suggests using two different books to cover both geometrical and algebraic perspectives on group theory, recommending Armstrong's "Groups and Symmetry" and Pinter's "A Book on Abstract Algebra." They also mention alternatives if Pinter is too basic.
  • Another participant shares their son's experience with Jacobson's books, highlighting their rigor and the importance of good problems for self-study.

Areas of Agreement / Disagreement

Participants express a variety of preferences for learning resources and approaches, indicating that there is no consensus on a single best method or source for studying group theory. The discussion reflects multiple competing views on the appropriate materials and strategies for learning.

Contextual Notes

Participants note that the choice of resources may depend on individual learning styles and goals, as well as the specific applications of group theory being pursued. There is also mention of the distinction between discrete and continuous groups, which may influence the selection of materials.

mertcan
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Hi, I saw that group theory is a significant asset for some physics, and math topics. I had some fundamental knowledge, but I am really keen on learning group theory deeply , so Is there a nice source( video links, books... whatever comes to your mind ) to leap further in this topic remarkably?
 
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Some basic courses, such as abstract algebra, may be useful. You can find this fundamental courses in most university. At least I know there exist some in MIT's school website.
 
Did you mention the attached screenshot? ıf yes, I have glanced at them , and have seen that the notes are so short, I mean Are this kind of short notes related to a significant topic like a group theory sufficient? I really wonder
 

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Is there anyone who wants to suggest some different and detailed sources?
 
mertcan said:
Is there anyone who wants to suggest some different and detailed sources?

There are many diferent and detailed sources out there. Everything depends on your personality.

For example, why do you want to learn group theory? Because you want to know its mathematics inside out, or simply because you want to understand representation theory in QFT? The two goals are very distinct. There are many other reasons to learn group theory, for example, if you want to understand groups in chemistry, then a pure math book won't be good for you.

What is your knowledge level in set theory and proofs? Do you know the basics? Are you a bit proficient in it?

Are you interested in discrete (finite) groups and symmetries, or in continuous groups and symmetries?

Do you want a pure math book with rigorous constructions and proofs of every statement, or is a book for physicists that is at places nonrigorous and more intuitive good enough for you.
 
micromass said:
There are many diferent and detailed sources out there. Everything depends on your personality.

For example, why do you want to learn group theory? Because you want to know its mathematics inside out, or simply because you want to understand representation theory in QFT? The two goals are very distinct. There are many other reasons to learn group theory, for example, if you want to understand groups in chemistry, then a pure math book won't be good for you.

What is your knowledge level in set theory and proofs? Do you know the basics? Are you a bit proficient in it?

Are you interested in discrete (finite) groups and symmetries, or in continuous groups and symmetries?

Do you want a pure math book with rigorous constructions and proofs of every statement, or is a book for physicists that is at places nonrigorous and more intuitive good enough for you.

Pure math book with rigorous constructions and proofs of every statement definitely covers my needs, besides I really want to both know its mathematics inside out and understand representation theory in QFT. I consider that I need probably 2 different kind of sources, books ( for QFT, and for ıt's pure mathematics ). Also I will be so pleased if such a nice sources are shared with me...
 
Brian Halls' book is nice. Georgi is nice if you want more emphasis on physical applications instead of math.rigor. Also, Zee has written a book recently on group theory, but I'm not familiar with that one.
 
mertcan said:
Pure math book with rigorous constructions and proofs of every statement definitely covers my needs, besides I really want to both know its mathematics inside out and understand representation theory in QFT. I consider that I need probably 2 different kind of sources, books ( for QFT, and for ıt's pure mathematics ). Also I will be so pleased if such a nice sources are shared with me...

OK cool. I'm going to advise that you work through TWO books for group theory. Why is this? Because I feel there are two ways of approaching the subject. One way is geometrical, where the important groups arise from geometry. Another approach is algebraically, where we just see a group as an algebraic object. Most books give one perspective and a bit on the other, but usually they're not comprehensive on both. I suggest that you read both books together.

So first for the geometrical perspective, there is Armstrong "Groups and Symmetry" https://www.amazon.com/dp/0387966757/?tag=pfamazon01-20 Very good book. Gives very nice insights. Lacks a bit of rigor though and lacks exercises (but that's why I also suggest to do a second book that makes up for this).

Then for the algebraic part, there's Pinter's "A book on abstract algebra" https://www.amazon.com/dp/0486474178/?tag=pfamazon01-20 Very cheap but good book. Covers quite a lot of group theory and also other kinds of algebra that you might not be interested in.

If you find Pinter a bit too easy for you, then I would go for: https://www.amazon.com/dp/1482245523/?tag=pfamazon01-20 This does a lot of algebra. You can read the chapters on group theory independently if you wish. It covers more than Pinter but it is still introductory. It has very very good problems (but so does Pinter).
 
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Ok, guys I am going to endeavour to dig lots of things out of these books :)) Thanks...
 
  • #10
mertcan said:
Ok, guys I am going to endeavour to dig lots of things out of these books :)) Thanks...

Note that haushofer's reply is a good one. Brian Hall's book on Lie groups is one of my favorite math books out there. But the books I listed and Hall are two different topics. My books are more about discrete (finite) groups, while Hall deals with the continuous situation. Now, the continuous situation is exactly the one you'll need for QFT, but I do advise doing the finite group situation first because it might give you some intuition you otherwise might lack.
 
  • #11
mertcan said:
Pure math book with rigorous constructions and proofs of every statement definitely covers my needs...
My son self studied abstract algebra. He's a pure mathematics guy who lives for proofs and absolute rigor. After a couple of tries with other books, he ended up working his way through Jacobson's https://www.amazon.com/dp/0486471896/?tag=pfamazon01-20 and https://www.amazon.com/dp/048647187X/?tag=pfamazon01-20. Apart from being quite excellent, they are Dover editions and therefore cheap.

Note that for self-study it is imperative that a book has good problems.
 
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