Tips on studying the classification of groups

Question

This may sound like a complaint (it is), but I feel like I am in need of some advice.

Honestly I am kind of overwhelmed. I just got done studying the Sylow Theorems (Chapter 8 in Stephen Lovett's Abstract Algebra), and right now am getting into the solvable groups, Index Theorem, Simplicity of PSL$_n(F)$ and of $A_n$ for $n>5$, and semidirect products.

The notations are getting confusing, and I do not get the motivation behind some theorems. Is there anything I should keep in mind while studying these topics? Any advice from someone who has walked this path would be helpful right now.

Answer 1

Judging from the content of the post, I think the title is slightly inproper, as you are simply learning the basics of group theory, not the classification of groups.

A good motivation for learning group theory, especially some of the stuffs you mentioned in your post, is Galois theory. There are many interesting examples/questions here:

• The classical compass and straightedge construction problems: impossibility of "angle trisection" and "doubling the cube";
• Construction of regular polygons;
• Solving polynomial equations by radicals;
• etc.

Try to read some introductory materials on Galois theory, and you'll probably get a better understanding of the importance of the corresponding notions in group theory. For example, why is a "solvable group" called "solvable"? Because it corresponds to the solvability of a polynomial equation by radicals...

Another interesting application of group theory is Burnside's lemma, which helps with many combinatorial counting problems.

It's true that learning without motivation is sometimes depressing. Having some application in mind will be much better.