Talk:Coset

message: 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)
thread_title:  picture 

message: 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)
thread_title:  Coset multiplication 

message: 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)
replies: text: 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)

text: 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)

text: 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)
thread_title:  Unfriendly? 

message: 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)
thread_title: Unfriendly, revisited

message: 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 \le S_3 \   as the first example instead.Hawthorn (talk) 00:59, 30 June 2009 (UTC)
thread_title:  Example 

message: 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)
thread_title:  Definition wrong, contradicts examples 

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thread_title:  External links modified 

message: Fourier Transform can be defined on cosets. I was hoping someone can expand on this. but  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)
replies: text: I mean whether you are able to find the Fourier Transform formula on coset. Jackzhp (talk) 03:05, 24 October 2018 (UTC)

text: can I expect you to add the Fourier Transform on cosets? Jackzhp (talk) 13:04, 3 November 2018 (UTC)
replies: text: 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)
thread_title: Fourier Analysis on Cosets

message: 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)
replies: text: 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)
replies: text: 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)
replies: text: 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)
replies: text: It might be succinct, but I can't understand it,    Comfr (talk) 03:54, 22 May 2020 (UTC)
replies: text: 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)
replies: text: Yes, Wcherowi made the article understandable.    Comfr (talk) 18:17, 23 May 2020 (UTC)
thread_title: Formal definition incomplete

message: 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)
thread_title: Semantics

message: We note that ba=a^{-1}b \Leftrightarrow ba=a^2b \Leftrightarrow aba=b 
 
b*ab=(ba)b=(a^2b)b=a^2 
b*a^2b=(ba)ab=(a^2b)ab=a^2(bab)=a^2a^2=a
ab*ab = (aba)b= (b)b=I 
ab*a^2b=(aba)ab=b(ab)=(ba)b=(a^2b)b=a^2
a^2b*x=a(abx)
KharanteDeux (talk) 16:58, 31 May 2021 (UTC)
thread_title:  Definition/First example