1) It says that some exponent rules hold, and others do not, for example (ab)^n may not equal a^n*b^n. But then we have theorem 7.7 which says that a^m*a^n=a^(m+n), and (a^m)^n=a^(mn). So is it safe to say that exponent rules don't really hold when the bases are different, but they do when the bases are the same? And also, the operation in the group could be addition, so when you say a^3, it really just means add a to itself 3 times, right? I don't really understand theorem 7.8, and corollary 7.9.
2) I think groups are really cool because they have all of these properties that I didn't consider and they are a little bit more abstract and challenging for my mind to wrap around than rings. It is a good stretch.
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