Learn Group Theory: Sources & Resources

In summary: I think it's a bit better if you read the whole thing from start to finish.Both of these books are excellent and will give you a good grounding in group theory. After reading them, I recommend doing some exercises from each book.
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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.
 
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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?
 
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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.
 
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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...
 
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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.
 
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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...
 
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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.
 
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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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1. What is group theory?

Group theory is a branch of mathematics that studies the properties and structures of groups. A group is a set of elements with an operation that satisfies certain axioms, such as closure, associativity, identity, and inverse.

2. Why is group theory important?

Group theory has wide applications in various fields, including physics, chemistry, computer science, and cryptography. It helps us understand the symmetries and patterns in nature and provides a powerful tool for solving problems in these fields.

3. What are some sources for learning group theory?

Some popular textbooks for learning group theory include "Abstract Algebra" by David S. Dummit and Richard M. Foote, "Algebra" by Michael Artin, and "Group Theory" by W.R. Scott. Online resources such as Khan Academy, MIT OpenCourseWare, and YouTube channels like MathTheBeautiful also offer free lectures and tutorials on group theory.

4. Are there any prerequisites for learning group theory?

A basic understanding of algebra, including concepts such as sets, functions, and equations, is necessary for learning group theory. Some familiarity with linear algebra and abstract algebra may also be helpful.

5. How can I apply group theory in my research or work?

Group theory has diverse applications in different fields, so it depends on your specific area of interest or expertise. For example, in physics, group theory is used to study symmetries in particle physics and quantum mechanics. In chemistry, it is applied to understand molecular structures and reactions. In computer science, group theory is used in cryptography to design secure algorithms.

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