Talk:Coset

Systems of common left/right coset representatives

See the discussion in Talk:four color theorem.

coset space

Need a discussion of the notation G/H (even in the non-normal case) and similarly for H\G. Dmharvey 20:07, 18 April 2006 (UTC)

This notation is not completely standard and you don't find it in most introductory modern algebra texts. I am not in favour of introducing additional notation unless it is needed and assists with the exposition. In this case I just don't see the need. It isn't as if we are going to have to repeatedly refer to the set of cosets and hence need a space saving notation for it. For an encyclopedia entry I think it makes sense to try to keep notation to a minimum. Hence I vote no Hawthorn 05:17, 30 May 2007 (UTC)

Umm, isn't it standard among anyone who would use H\G or more than one action? Well, it does need to be discussed, if only to list which areas use it which way. Wikipedia isn't a text book with it's own consistant notation. Instead Wikipedia does need to concisely explain all common notations. The point is that researchers in other areas and graduate students come here to figure out what is going on with the basics in other articles that they are reading. 134.157.19.52 (talk) 10:57, 7 April 2008 (UTC)

"so they're probably not in the public domain"

That doesn't matter. Just recreate the picture yourself and it will be-- particularly if you tweak the picture to conform to the objections you mentioned.

If you paint a copy of the Mona Lisa yourself, Leonardo can't barge in and steal it, because he doesn't own it. You do. You just can't sell it as "the Mona Lisa".

HelviticaBold 00:26, 15 April 2012 (UTC)

picture

I have a picture that illustrates coset, among other things, let me know how I can tweak it so it fits with the terminology of the current article--Cronholm144 09:34, 31 May 2007 (UTC)

I agree. Some pictures would be helpful

--Farleyknight (talk) 07:13, 22 February 2009 (UTC)

These pictures look like they're taken from Artin's Algebra book (so they're probably not in the public domain). Also, that picture isn't terrible helpful: consider the C/U_1 (complex numbers mod complex numbers with length 1). We know U_1 is normal because it's the kernel of the map x --> |x| from the non-zero complex numbers to the positive reals. The cosets are circles about the origin, not lumps.

Again, consider a one dimensional subspace of the real plane. The cosets are lines parallel to the one dimensional subspace. Here, operating on the cosets is translation. —Preceding unsigned comment added by 66.67.62.188 (talk) 01:53, 14 July 2009 (UTC)

Coset multiplication

I can also draw an image for coset multiplication, but the article barely mentions it in the quotient groups sub section. Let me know if you want the image--Cronholm144 10:09, 31 May 2007 (UTC)

N is the kernel, the rest are cosets

In the introduction I don't get a general Idea of what a coset is. So it is actually difficult for me, reading through the article, to understand what a coset.

Unfriendly?

For someone coming in cold, this is probably confusing. If they have recently come to understand sets and groups, the notation gH could be obscure; and

{gh : h an element of H }

more so. An educated guess would be that gH means "apply the group operation in turn to g and every member of H", yielding a subgroup of G as the result so that, if the members of H are h¹, h², h³, etc, the coset is gh¹, gh², gh³, etc.

If I've got this correct (can anyone confirm?), a textual description preceding the more formal exposition would be helpful. Fishiface (talk) 18:50, 5 November 2008 (UTC)

That's almost right. But gH isn't a subgroup unless $g\in H$, in which case gH = H. To see why that's the case, note that the cosets partition the group into (disjoint) subsets and the identity belongs to H. Can you think of a better way to phrase what's there? I'm coming at it from having already studied this, so it's hard to spot unclear bits. Rswarbrick (talk) 18:04, 23 December 2008 (UTC)

Another unfriendly bit is the sentence (in the General Properties section):

gH is an element of H if and only if g is an element of H, since as H is a subgroup, it must be closed and must contain the identity.

Although the bit to the left of "since" is true, the bit to the right of "since" doesn't explain why.70.179.92.117 (talk) 03:35, 16 December 2010 (UTC)

Since groups are divisible, the right coset contains the identity iff 1/g is in H, and if 1/g is in a subgroup, 1/g must be divisible, so g must be in that subgroup. ᛭ LokiClock (talk) 01:35, 15 April 2012 (UTC)

Honestly, I find this article bad redacted when compared to other articles about set theory and abstract algebra. The difinition is not well written, it just slaps the mathematical definition without saying what it is first as in all other articles.

Unfriendly, revisited

Re: fishface's objection, above. I am about at his level (or somewhat less). I understand the meaning of the symbols, but I still don't know what a coset is or what it's good for. The first paragraph doesn't give a definition of "coset"; it makes the distinction between left and right cosets.

Note that this is supposed to be for actual human people who know what group theory is and also want to know what a coset is and how it's used. So it is not sufficient to merely prepend the article with the BNF-like definition: "A coset is defined as {x: "x is a left coset" v "x is a right coset"}, and then go on to distinguish between the two.

Thank you. [curtsies...]

HelviticaBold 00:41, 15 April 2012 (UTC)

Example

All examples currently are of cosets in an abelian group. Abelian groups are atypical and use the alternative additive notation. I think the first example should be typical and use the more common multiplicative notation. How about using $\ \langle (12)\rangle \leq S_{3}\$ as the first example instead.Hawthorn (talk) 00:59, 30 June 2009 (UTC)

Definition wrong, contradicts examples

The definition clearly states that cosets are only considered for elements $g\in G$ that are NOT elements of $H$. This is not correct. In fact, the examples later in the article clearly consider elements of $H$ to generate the coset $H$ itself. (See the $G=\{-1,1\}$ example, even.) What's up with that?

I went ahead and changed the definition to remove that condition on $g\notin H$. Please tell me if I'm just being dumb.

Bwsulliv (talk) 01:27, 25 September 2012 (UTC)

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Fourier Analysis on Cosets

Fourier Transform can be defined on cosets. I was hoping someone can expand on this. but Wcherowi (talk · contribs) revert my editting. He asked "Many things can be defined on cosets. What makes this special?". there are 4 other items inside the Applications section, what make those special? Isn't Fourier transform special enough? Are you able to do Fourier Transform on coset space? Give you 10 days, if you are able to do so, then it is not special, if you are not able to do, I guess you should agree it is very sepcial. Jackzhp (talk) 03:02, 24 October 2018 (UTC)

I mean whether you are able to find the Fourier Transform formula on coset. Jackzhp (talk) 03:05, 24 October 2018 (UTC)

To editor Wcherowi: can I expect you to add the Fourier Transform on cosets? Jackzhp (talk) 13:04, 3 November 2018 (UTC)

As you have not responded to my direct question, I see no reason to do so. Your complaint about other items and rather childish challenge do not constitute a response. It seems clear to me that you have no source that would justify the inclusion of this material and asking other editors to do your work for you is generally frowned upon. --Bill Cherowitzo (talk) 17:56, 3 November 2018 (UTC)

Unfriendly, yet again

The lead paragraphs of this article leave something to be desired for the casual reader. True, the definitions are all present and correct, but the visitor to this page who is unfamiliar with the topic will have a hard time putting the pieces together to get an overall picture (see some comments above and some failed attempts to improve the page). I propose changing the introductory paragraphs in a slightly unorthodox way by talking about how the cosets (right or left) form a partition of the underlying set of G, without bringing up the equivalence relation that defines them. This would give the novice reader a visual interpretation that will help clarify the definitions that follow and show why they are needed. The equivalence relation can be introduced later after the reader has gained some insight (it is a more abstract concept than a partition). I say that this approach is slightly unorthodox since all the texts that I am familiar with bring up the partition as a consequence of the equivalence relation (which it is), but I am proposing to divorce the two ideas, at least in the introduction. If I get some support for this I'll come up with some proposed language.--Bill Cherowitzo (talk) 18:06, 18 May 2020 (UTC)

I support fixing the article. I will say more after check some references. Thank you. Comfr (talk) 02:21, 19 May 2020 (UTC)

The introduction should plainly state that each element of G is in exactly one and only one coset, however multiple elements of G can occupy the same coset. (This is like saying that everyone has a street address, but that multiple persons can share an address.) Eventually the article says, "Thus every element of G belongs to exactly one left coset of the subgroup H," but this critical statement appears too late.

The use of Set-builder notation comes too early in the article.

The article should also explain how to determine which cosets exist, and which elements belong to which coset. In the table below, I had to do 16 calculations to determine that only 4 actual cosets existed. Only after creating and analyzing this table did I finally understand the definition of a coset.

The lead should contain a summary, with another section containing definitions.

I agree that cosets partition a group, but I would be careful to say what partition means.

Thank you for recognizing the unfriendly introduction. Comfr (talk) 01:35, 20 May 2020 (UTC)

The introduction is now very friendly and beautiful. Thanks for the time and effort you put into this article. Comfr (talk) 21:03, 21 May 2020 (UTC)

Formal definition incomplete

A coset is actually a set of sets. This compounding makes attempts at definition confusing.

For each element g of G, the product gH is a set of |H| elements. There are |G| such products, and the set of those products is the coset, having only [G:H] elements, because by definition, sets do not have duplicate elements.

As written, the definition does not tell how |G| products forms exactly one coset.

I changed the plural cosets to singular to avoid some confusion, but at this point, I do not know how to fix the definition. Comfr (talk) 22:37, 21 May 2020 (UTC)

More at Transversal_(combinatorics)#Examples. Comfr (talk) 22:43, 21 May 2020 (UTC)

@Comfr: This seems unnecessarily convoluted. A coset is simply an equivalence class under the equivalence relation x~y if and only if $x^{-1}y\in H$. There's no need to appeal to sets of sets even though that's possible.--Jasper Deng (talk) 22:56, 21 May 2020 (UTC)

I have just now added a sentence that might clarify this for you. Thinking of a coset as a set of sets is just not productive. I think you are misreading the transversal article's statement. The set of all cosets (of one type or the other) form a partition of G. A transversal is a set obtained by picking one element from each of the cosets. The elements of the transversal are called coset representatives in this case.--Bill Cherowitzo (talk) 23:28, 21 May 2020 (UTC)

Another way to define a coset, perhaps more directly, and still without reference to a set of sets, is that the cosets of H in G are the orbits of the group action of H on G by $(g,h)\to gh$ or $(g,h)\to hg$ depending on left or right. For if two elements of G are in the same coset, their "quotient" must be in H, and if not, then obviously by definition, there is no element of H that can multiply by one of those members of G to give the other. Nice and succinct.---Jasper Deng (talk) 00:58, 22 May 2020 (UTC)

It might be succinct, but I can't understand it, Comfr (talk) 03:54, 22 May 2020 (UTC)

If you can't understand it, then quite honestly I advise you to study this field a little more before suggesting a new formal definition. It's not to say you don't have a valid point, but Wcherowi's addition should address any confusion.--Jasper Deng (talk) 09:19, 22 May 2020 (UTC)

Yes, Wcherowi made the article understandable. Comfr (talk) 18:17, 23 May 2020 (UTC)

Semantics

The article says, "given an element g of a group G and a subgroup H of G...," which implies there should be one coset for each value of g. However, some members of G share the same coset, so there are fewer cosets than members of G. The definition should make clear exactly what a coset is. Consider the following:

Cosets of H in G

Value of element g	Left coset	Right coset
0	{0+0,0+4} = {0,4}	{0+0,4+0} = {0,4}
1	{1+0,1+4} = {1,5}	{0+1,4+1} = {1,5}
2	{2+0,2+4} = {2,6}	{0+2,4+2} = {2,6}
3	{3+0,3+4} = {3,7}	{0+3,4+3} = {3,7}
4	{4+0,4+4} = {4,0} = {0,4}	{0+4,4+4} = {4,0} = {0,4}
5	{5+0,5+4} = {5,1} = {1,5}	{0+5,4+5} = {5,1} = {1,5}
6	{6+0,6+4} = {6,2} = {2,6}	{0+6,4+6} = {6,2} = {2,6}
7	{7+0,7+4} = {7,3} = {3,7}	{0+7,4+7} = {7,3} = {3,7}

Comfr (talk) 02:13, 19 May 2020 (UTC)

Definition/First example

We note that $ba=a^{-1}b\Leftrightarrow ba=a^{2}b\Leftrightarrow aba=b$

$b*ab=(ba)b=(a^{2}b)b=a^{2}$
$b*a^{2}b=(ba)ab=(a^{2}b)ab=a^{2}(bab)=a^{2}a^{2}=a$

$ab*ab=(aba)b=(b)b=I$
$ab*a^{2}b=(aba)ab=b(ab)=(ba)b=(a^{2}b)b=a^{2}$

$a^{2}b*x=a(abx)$

KharanteDeux (talk) 16:58, 31 May 2021 (UTC)