The most difficult part of this section for me to understand was actually one of the examples. The second example on page 218 was confusing because they did the arithmetic for the example but didn't do any of the verification that G/N is cyclic and of order 4. I think that I am confused as to what a cyclic quotient group looks like and how to use a multiplication table to determine that. If I could get to the determination that it had an element of order four then I would know that it was cyclic and that it was isomorphic to Z4. But I can't figure out how to tell from the table that something has order 4.
I thought the most interesting part of the section was the last theorem where it is shown that if G/C is cyclic then G is abelian. Do we know in what cases G/C is cyclic? Does it happen often? It would be nice to look at examples where this theorem is true. I would also like to look at examples where G is abelian but G/C is not cyclic.
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