7.9, due on November 9

I didn't understand the whole second half of the section starting with transpositions. I guess what was the source of my confusion was starting with how (12)(12)=(1). I think I might be just not understanding the notation. After that, I couldn't follow the whole odd or even number of transpositions proofs and the last theorem 7.51 was the most confusing. I did not understand the definition of An and the proof that followed.

I thought the most interesting part of the section was that we developed the notation for symmetric groups to simplify problems. It seems like a really clever and simple solution to writing the long form of elements in a symmetric group. I also thought it was interesting that theorem 7.51 was proved using the First Isomorphism Theorem. I want to know how this theorem is proved when you learn about group theory before ring theory and don't have the First Isomorphism Theorem to use.