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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. (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)


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)

Group homomorphism.svg

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 (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.


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 , 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. (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)

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)


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 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)