I am an undergraduate and I have studied basic things like groups,subgroups,cosets and Lagrange's theorem,homomorphism,isomorphism,direct product etc. and now I want to self study the topic class equation and Sylow theorems.I have some queries regarding the same:
$1.$What are the prerequisites of Sylow theorems and class equation.
$2.$What are the main topics that I must cover for first reading to get a complete overview of those topic.(I mean the topic headings)
$3.$What is a suitable book to study class equation and Sylow theorems for a first time reader and a beginner.
$4.$How should I apporach these topics,should I read the formal theorems and develop intuition afterwards or should I study examples first and then go to the general theorems.
Best Answer
Essential for understanding the class equation aswell as the Sylow Theorems is a decent knowledge of group actions. While the former (the class equation) is formulated in the language of groups actions, the latter (the Sylow Theorems) are commonly proved using ideas and techniques based on or around group actions.
Study group actions first and understand on the one hand which particular subgroups and subsets are of importance and on the other hand how the basic theorems (orbit-stabilizer theorem, Burnside Lemma) work and can be applied in different situations. Try to develop an intution what these are all about (for myself it helped to look at it from a categorical point of view, as Emily Riehl does, but this might be a bit to much for a first read into group actions).
Two references come immediately to my mind: T. Judson "Abstract Algebra, Theory and Applications" and D. Dummit, R. Foote "Abstract Algebra". The latter is a standard reference and the first is freely available and can be found here for instance. From what I recall, the first is especially accessible. In addition, the Sylow Theorems are treated right after the chapter dealing with group actions but still seperated, which might help for clarification.
Read the statement and proof of the Sylow Theorems. While going trough the proof pay attention how the the concepts of groups actions you learned earlier come in handy. Applications of Sylow's Theorems exemplify how useful it is to decompose the order of a group into its prime factors. You can show interesting things, like every group of order $45$ is abelian, which at a first glance has nothing to do with group actions at all. Work through some of these examples to gain an intuition for the Sylow Theorems and what they can give you in return when applied under suitable circumstances.