Learn Group Theory: Sources & Resources

[mertcan]

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?

 

[tommyxu3]

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.

 

[mertcan]

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

 

[mertcan]

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

 

[micromass]

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.

 

[mertcan]

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...

 

[haushofer]

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.

 

[micromass]

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).

 

[mertcan]

Ok, guys I am going to endeavour to dig lots of things out of these books Thanks...

 

[micromass]

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.

 

[IGU]

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.