By far these were the most confusing sections of the class. I don't understand what is going on with the Sylow second theorem. I don't even know what they mean by P=x^-1Kx because I thought that doing x^-1Kx was used to determine if K is normal in G. What would it look like to have two sylow p-subgroups that were equal in this way? But my level of confusion really grew in section 8.4. I can read theorems and think I understand them, but then reading the proofs really showed my that I did not. I was so confused by the use of the conjugacy classes as used in the proofs. I also was getting mixed up and confused by the many uses of Lagrange's theorem. I need to reread these sections again and I need to go through the proofs in class really slowly.
I thought that the most interesting part of the section was talking about conjugacy classes and the centralizer. Throughout the semester as I have looked up proofs and information for clarification, conjugacy and centralizer have come up really frequently. People are always using them as the easiest way to prove things that we learned earlier on (which is obviously unhelpful at the time because we hadn't learned it). I am happy to finally be talking about them because (once I funny understand them) I think it will make a lot of aspects of group theory more clear - or at least I'll be able to understand some more of the online discussions and proofs of group theory.
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