Talk:Commutator subgroup

Notation

As pointed out above, the notation $G^{1}$ is unusual. If there are no objections, I intend to change it to $[G,G]$ (because the alternative $G'$ is too easily misread as $G$ in some fonts). --Zundark 12:49, 15 February 2006 (UTC)

The notation $G^{(n)}$ is nice in some respects because the condition for a group to be solvable is that there exists some n such that $G^{(n)}=\{e\}$, echoing ideas of differentiation somehow. But you are right, it isn't all that common. I much prefer $G'$, but [G, G] should be okay too, though it might look a bit cumbersome in the article. — Preceding unsigned comment added by Dysprosia (talk • contribs) 10:20, 16 February 2006 (UTC)

'Smallest' normal subgp?

This article seems to claim a total order on normal subgps. This is probably a mistake but I wanted to check first.

Isaac (talk) 01:28, 19 December 2007 (UTC)

The article does not (intend to) claim that the normal subgroups are totally ordered, merely that a certain partially ordered subset of normal subgroups has a unique minimal element. The commutator subgroup is the intersection of all normal subgroups with abelian quotient and is itself such a normal subgroup, so it is the unique minimal element of the set of normal subgroups with abelian quotients, where the order is the partial order of set-wise inclusion. It is reasonably common in mathematics to use "smallest" for partial orders to be sort of a two-part assertion: (1) it is *a* minimal element, that is, there are no elements less than it in the partial order, and (2) it is the only minimal element, and even more, every element is greater than it in the partial order. A poset need not have a smallest element, but it can. Infimum or meet are other common words used in partial orders and lattices. The poset article uses the term smallest (with the sort of required qualifier, "if it exists"), and the lattice article uses "least" and "infimum" (where part of being a lattice is being a poset where smallest elements *do* exist; the lattice of normal subgroups is a lattice for instance). JackSchmidt (talk) 03:45, 19 December 2007 (UTC)

Last line of Definition

"The commutator subgroup is a fully characteristic subgroup: it is closed under all endomorphisms of the group, which is stronger than normal."

Can someone clarify what is meant by the last clause 'which is stronger than normal'. Does it mean "The condition of being closed under all endomorphisms is a stronger condition than obtains for normal groups" ? As it is it kind of hangs in the air and isn't clear (to me anyway) Thx. Zero sharp (talk) 05:42, 10 April 2008 (UTC)

I've reworded it. --Zundark (talk) 07:45, 10 April 2008 (UTC)

Some cleaning up

I made some changes in the article. Previously, the facts that the commutator subgroup was normal and moreover fully characteristic came up three times in the article, each time saying things slightly differently. I consolidated this information into one place farther up in the article. I also wrote out some simple "commutator identities" so as to make these properties as clear as possible. I noticed that there was no discussion of "commutators" per se so I added a section on that.

I reiterated the assertion of the two groups of order 96 in which the subset of commutators is not a subgroup, but in fact this was news to me (not that I disbelieve it): a reference is really needed here.

I also removed the passage about the (supposed) universal property of the commutator subgroup, together with its rather awkward footnote. I think this material was actually incorrect: a universal mapping property is something that defines an object up to canonical isomorphism in a certain category. The fact that it is only defined up to canonical isomorphism is an important feature of the definition. But there's no category here; we're just talking about the subgroups of a fixed group [true, you can view the lattice of subgroups as a category in which the morphisms are the inclusion maps, but that's a contrivance in this context] and indeed the commutator subgroup is truly unique, a clue that there is no [nontrivial] universal mapping property here.

On the other hand, the universal mapping property of the abelianization ought to be spelled out, although perhaps in a different article. Plclark (talk) 11:39, 3 July 2008 (UTC)

I noticed that JackSchmidt made some changes in my revisions. Thanks for this -- almost all the changes are clear improvements. Just a couple of thoughts:

1) This is picky, but I think that always -- and especially in the more formal context of an encylopedia, definitions should come before they are used, not after: so the conjugation notation should not be used and then defined followed by a "where..."

2) I had defined the commutator as $ghg^{-1}h^{-1}$ and this was changed back to having the inverses first. Of course there is no mathematical difference here. But in my experience, the convention I used seems more common and is a little more sensible (why throw in inverses? we could also define the commutator as $hgh^{-1}g^{-1}$ and that would be mathematically okay too, but shouldn't the most direct and transparent notation be used?). The article on commutators suggests that the "inverse first" convention is preferred by group theorists. Is this (still) true? I would be interested to hear from a group theorist on this.

Also, the section on abelianization should be expanded a bit, since it seems not to appear anywhere else on wikipedia. Does anyone else think that the use of terminology like "reflective subcategory" and "reflector" is more heavy-handed than necessary?

Thanks again -- it's a very positive experience to have one's improvements improved upon.

Plclark (talk) 21:52, 3 July 2008 (UTC)

Thanks for the explanations on the talk page. Today has been hectic, and I hadn't noticed the clear explanations here (luckily I don't think I messed up anything you explained). I definitely liked your reorganization. Looking at the diffs I was at first appalled you had deleted so much good material, but then when I looked at the article it was all still there! Magic? No just repetitive old version. The section on commutators seems like a very good idea. It has just the right amount of detail before sending the reader to commutator or down to the next section for the commutator subgroup proper.

I'll see if I can find a paper reference for the 96 thing. It is in a few textbooks (Dummit&Foote's Algebra, Isaacs's Character Theory, probably many others). Sadly it takes less than a second to search the small groups library in GAP, so it may not ever have been published as a "result". There was also some interesting thing about rearranging words and the relation to commutator subgroups and commutators, but I've forgotten the authors (I think it was a series of papers).

I'm never too fond of using categorical language to describe simple ideas in group theory. Even Glauberman's text on finite groups says that "conjugation functors" (a technical device that behaves well with conjugation and normalizers) are just called that, though you could make a category if you wanted.

So yes, I think if "reflective subcategory" is needed to understand a point in the article that is not intended for explaining category theory, then I think it is too heavy-handed. On the other hand, there is no reason not to have some explanations of category theory in this article, just like explaining a bit of the commutator subgroup in the commutator article. In other words, it is fine as "bonus material". I actually feel excluding these points of views (even if they are heavy handed) is a violation of WP:NPOV, but that policy also says they need only be given their due weight based on how prevalent that point of view is.

To answer your specific concerns about my edits: (1) I'm fine with moving the definition up. I believe it is done to your liking in the commutator article. I tend to think it is fine to have the definition in the same sentence. (2) This has always struck me as odd. More or less every group theorist uses gh = hg[g,h] = h g^h, the only exceptions being a few who came over from topology. However, really a significant portion of all other mathematicians (rarely use it at all, but when they do they) use gh = [g,h]hg. In the commutator article, both standards are discussed. In the commutator subgroup article basically it doesn't matter which way you do it, so I just try to keep it from changing.

So, I am fine with any fixes to 1 (sorry, I hadn't realized I broke it). I lean heavily towards keeping the "English" (and German) convention for gh=hg[g,h], but lean even more heavily towards trying not to mention it at all in this article. Definitely need more sources (in all the math articles!). You have my support for making all category theory stuff into "extras", rather than the main presentation, but try to keep some (correct) mention of it. JackSchmidt (talk) 22:38, 3 July 2008 (UTC)

Short description

The short description of this article is currently "Smallest normal subgroup by which the quotient is commutative." The definition of the commutator subgroup is "The subgroup generated by the set of all commutators." I propose changing the short description to the definition. What is currently in the short description is a consequence of the definition.—Anita5192 (talk) 20:38, 12 April 2020 (UTC)

For the record, the previous short description was "smallest normal subgroup such that the quotient of the parent by this subgroup is commutative". IMO, the present short description better fits the objectives short descriptions, as described in WP:Short description#Content. In fact "Duplication of information already in the title is to be avoided", so, it is better to avoid repeating the word "commutator". "The short description may be used for several purposes, including as a disambiguator in searches and as an annotation in outline articles." This implies that the short description is mainly used by readers for deciding whether they want to access the article. So, it is better to explain the reason for which the concept is useful than to provide the technical definition, specially when, as it is the case here, both are equivalent and this eqivalence is given in the second sentence of the lead. So the present formulation seems better. However, I am open for changing my opinion if strong arguments are given against my formulation.

By the way, you can see in Abelian group#See also an example of the use of the short description of this article, and of the redirect to one of its section D.Lazard (talk) 02:45, 13 April 2020 (UTC)