For this section of the book, I think that Lagrange's Theorem, Cayley's Theorem, and the first Isomorphism theorem are pretty important. Usually when theorems are named they are a big deal. I also think that other theorems will be really important too just because it is useful information to know in helping prove other things. For example, I feel like the theorems that give easier or faster ways to determine if something is a sub ring, or subgroup are really helpful. Also, the theorems that just are simple things to know to be able to use in other proofs, for example inverses are unique, or cancellation holds, or the identity is unique.
I expect to see questions about proving if something is an ideal, or a group. But I could also see things like find a maximal ideal, or prove something about the kernel, or show that this group is cyclic. I think there will also be questions about finding the cosets of an ideal or group. Also, like the last exam, I think there will be questions asking for examples of groups, ideals, subgroups, etc.
Something that I would like to discuss before the exam is that I really want to go over the main differences of the main topics we have covered in this unit. Another thing that I really struggle with is maximal ideals, cosets, and cyclic-generated groups. So I would really like to see examples of these things so that maybe some of my confusion could be cleared up.
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