Difficult: So in the first part of this section, the ideas are pretty parallel to those of rings. I understand homomorphisms, but the hardest part for me is finding the map f that is a homomorphism. It is easier for me to prove, but is there some kind of way to know what the map is? In all of the examples in the book, they choose their map, but this is something that is hard for me to do. Also, the idea of simple groups makes sense to me, but I'm just not sure how to go about testing if a group is simple. Is there a general way to do it? Also, I kind of got lost in the explanation of why simple groups are important. I didn't understand the point of them, I think I might have just gotten lost in the notation.
Connections: The First Isomorphism Theorem for groups is very similar to that for rings. I feel very confident in my ability to prove this theorem for rings, and I feel that I understand it pretty well. As a result, I think that I would be able to do this proof for groups as well. It follows the same general idea, but some of the notation is changed and some definitions would look different. Again, I am left wondering how they came up with the map for phi? How did they know to pick phi(Ka)=f(a)?
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