Talk:Quotient group

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About the Example of R/Z. Can you explain it better to me? R/Z is the set of cosets of Z. So we have: cosets={r+Z} for all r in R. This is very important, the cosets are parametrized by r, not by Z. In the case of r equal or bigger as one, we stay in the same coset, by taking the right integer z', r=x mod 1. So that's why we can restrict r being 0<=r<1. That is a representative of the coset so to say. The map r->e^{2\pi i r} is clearly a homomorphism, as addition in r is multiplication in S^1. f(R/Z)=S^1.

But I can be wrong in the argumentation. Feel free to point out the error.

The domain of the homomorphism is the set of cosets. The image of Z is 1, but Z is only one element of the set of cosets.--Patrick 11:35, 26 August 2005 (UTC)

Aha... :-) I mixed things up with Z and mod 1 (as being a representive) in the cosets. Not each coset has Z, only r=0. That all r+Z for fixed r are mapped to the same value under the homomorphism doesn't matter, as different cosets are characterized by different r.

[edit] The examples need help

As a non-mathematician with a mathematical bent (an engineer), I find the examples quite unhelpful as they currently stand. Can someone who understands them relate the examples to the concepts in the original article? The roots of unity especially seems to be on the verge of being very helpful, but stops just short of actually relating the coloured dots to the concepts introduced in the article. (Which dots are in G? Which are in N? Which are in G/N?) I find the other examples completely impenetrable. Thanks! --P3d0 19:12, 7 November 2005 (UTC)

All dots are in G, the red ones in N, while G/N is a set of three sets of 4 dots of the same color each.--Patrick 23:14, 7 November 2005 (UTC)

All dots are in G, that is in the first sentence in the example. Its subgroup is the red dots, that is second sentence. One has to think a bit what subgroup is that, but it is clear that this article is all about the normal subroup called N. The last sentence says that the three colors form a group, and it says that it is called the quotient group. All the info is there, just not as explicit as you wish. I made it now a bit more explicit. Oleg Alexandrov (talk) 00:38, 8 November 2005 (UTC)

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Nice job with the roots of unity example! Can the same be done to the others? --P3d0 05:44, 8 November 2005 (UTC)

I don't understand. That example seem to be the only one that was missing the added clarifications. JPD (talk) 09:52, 8 November 2005 (UTC)

Ok I used to think the first two examples were just one long example. Thanks to whomever added the bullets. But most of the examples (particularly the second one) are still baffling; if I understood them then perhaps I could explain what's necessary to make them understandable to non-experts, but unfortunately I am a non-expert. And only the roots of unity and the final example actually refer to the G and N used earlier. I'm sorry I can't be more helpful here. --P3d0 20:50, 8 November 2005 (UTC)

Come on, you are asking too much. When one says the group, that is meant to be G. When one says subgroup that is meant to be N. When one writes Z₄=Z/4Z one means that on the right hand side one has G over N. I think that is rather clear from context, and I would not want that G and N show up everywhere. Oleg Alexandrov (talk) 21:21, 8 November 2005 (UTC)

Can we at least conclude each example with a phrase in the form of "therefore foo / bar = baz" to tie everything together? --P3d0 14:38, 9 November 2005 (UTC)

Most of them have something like this somewhere, not necessarily at the end. I tried to make the point at the end of the second-last example a bit clear, and spelt out what the elements of the quotient group in the last group are in terms of cosets, but I can't see much else that needs fixing. Should the roots of unity picture have a caption? Apart from anything else, images without captions don't get properly right justified on my screen, and I don't know what to do about that. JPD (talk) 15:33, 9 November 2005 (UTC)

Ok I'm going to go ahead and add what I would like to see in there. Someone fix what I add if it's wrong. --P3d0 16:02, 9 November 2005 (UTC)