# Talk:Quotient group

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## Circle group

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.

## 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! --Doradus 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)

## Roots of unity

Nice job with the roots of unity example! Can the same be done to the others? --Doradus 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. --Doradus 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? --Doradus 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. --Doradus 16:02, 9 November 2005 (UTC)

## red x blue?

"the product of a red element with a blue element is red, ..."

Isn't red the identity element of the quotient group?

Aqm2241 18:07, 14 December 2006 (UTC)

Yes. I've fixed it. --Zundark 18:16, 14 December 2006 (UTC)

## Univers/Category

Don't we have a WP article on the category theory version of this concept? I can't seem to find it. linas 14:55, 28 February 2007 (UTC)

Dohh never mind, brain fuzz -- quotient category. linas 14:57, 28 February 2007 (UTC)

## Motivation Section

I added this section because I think it helps clarify the definition, but I am not sure this is the correct heading, or if it even belongs on this page as opposed to some other other page on general quotient structures. —The preceding unsigned comment was added by Padicgroup (talkcontribs) 15:36, August 20, 2007 (UTC).

## Topological groups?

Should quotient groups of topological groups by closed normal subgroups be added here? If so, should the quotient homogeneous space by a closed (not necessarily normal) subgroup be included here or at quotient space (topology)? silly rabbit (talk) 13:21, 14 May 2008 (UTC)

A short mention here of quotient "group objects", and the main explanation at topological group sounds fine to me. Coset spaces probably do not belong here. There is already a homogeneous space article, I think. JackSchmidt (talk) 13:30, 14 May 2008 (UTC)
That sounds reasonable. I think a link to homogeneous space would be sufficient. silly rabbit (talk) 13:33, 14 May 2008 (UTC)

## Quotient group definition(group or set)?

in this example provided for definition of quotient group

   G = {0, 1, 2, 3, 4, 5}.


Let

   N = {0, 3}. are the groups with operation addition modulo 6


the quotient group G/N = {{0,3},{1,4},{2,5}} what is the corresponding operation for this group??? how is it defined?

in the definition need proof that quotient group is a group49.137.185.48 (talk) 12:53, 9 April 2012 (UTC)sandeep

These questions are already answered clearly in the article. — Anita5192 (talk) 18:30, 9 April 2012 (UTC)
Yes they are answered in the first few lines itself,thank you.(talk) 02:49, 10 April 2012 (UTC)sandeep

## Technical article

The introduction doesn't say what a quotient group of a subgroup is. It doesn't say for any set G that's a group under any operation, ×, that the set of all cosets of a subgroup H is its quotient group if and only if multiplication of any 2 cosets of that subgroup can be defined in such a way that for any a and b in G, Ha × Hb = Hab. i.e. For any a, b, c, and d, if Ha = Hc and Hb = Hd, then Hab = Hcd. It doesn't state that H has a quotient group if and only if it's a normal subgroup. Also the information in the section Product of subsets of a group seems totally irrelevant and nothing to do with quotient groups and I think it should be replaced with information about coset multiplication, which is a different operation than subset multiplacation of those 2 cosets, that is, in this context, we define the product of 2 cosets of H to be another coset of H, not the set of all ordered pairs of an element of one coset and an element of the other coset as in Product of group subsets. Source: Pinter, Charles C. A BOOK OF ABSTRACT ALGEBRA 2nd ed Page 148 Blackbombchu (talk) 23:54, 10 March 2014 (UTC)

## Vector Space example

As is, the following example is bordering on original research so I'll stick it here until I or someone else can find an appropriate reference.

A vector space V is an abelian group under the usual vector addition (the zero vector 0 is the identity of the group, -v is the inverse of v, etc.). It is easy to see that a linear subspace N is then a subgroup. Since all subgroups of abelian groups are normal subgroups, one can define the quotient group Q = V/N = { v + N : v \in V }. In analysis, Q is called a quotient space and its elements are equivalence classes of vectors, where two vectors are equivalent if they differ by an element of N. The Lebesgue space L1 is an example; it is a quotient space constructed by taking the space of integrable functions and "quotienting-out" the subspace of functions which are zero almost everywhere. Usernames are difficult (talk) 17:04, 3 July 2014 (UTC)

Much of what you've written seems to be captured at Quotient space (linear algebra). I can't comment on Lebesgue spaces, though. —Quondum 01:40, 4 July 2014 (UTC)