Prove that is not cyclic.
How do I do this my book says nothing.
If was cyclic, then being the image of under the projection mapping would be cyclic, so it suffices to show that is not cyclic. To see this suppose that there was an isomorphism we see then so that is divisible by for every and so , but since this contradicts injectivity of .
To show that is not cyclic, it seems conceptually more straightforward to take an arbitrary number (say, in lowest terms) and say that it can never meet , where is some prime that is not a factor of . (For example, no multiples of will ever be .) So no single number can generate all of .
Sure, either way works.
Another good way to look at it! My method shows more generally that any divisible group is not cyclic.