Talk:Commutator subgroup

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Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. needs refs, try finding some here.--Cronholm144 22:47, 28 May 2007 (UTC)

I changed f(G') being a subset of H' by subgroup (since G' is a subgroup, not just the set of generators (it's the generated ("spanned") subgroup)) and thus f(G') is a subgroup of H. But f(G') is contained in the subgroup H', and thus taking the intersection H' \cap f(G') gives the subgroup f(G') drini ☎ 05:31, 31 May 2005 (UTC)

Contents 1 Title 2 Notation 3 'Smallest' normal subgp? 4 Last line of Definition

[edit] Title

Why is this "derived group" and not "derived subgroup"? Dysprosia 09:10, 13 February 2006 (UTC)

I think it would be better to call the article derived subgroup (or commutator subgroup). I also question the notation G¹. I don't think I've ever seen this notation before - usually G' or [G,G] is used, or sometimes (in the context of the derived series) G⁽¹⁾. --Zundark 12:54, 14 February 2006 (UTC)

I agree with you (derived subgroup should be fine, I think that is more common, but I haven't done a Google-check). Dysprosia 07:10, 15 February 2006 (UTC)

I've just tried Google (Web, Groups and Books), and also Zentralblatt. All four of these searches indicate that "commutator subgroup" is more significantly more common than "derived subgroup". (Some of these seaches get more hits for "derived group" than for "derived subgroup", but this could be due to the non-mathematical usages. I don't think there is any non-mathematical use of the expression "commutator subgroup".) So I suggest we move the article to commutator subgroup. --Zundark 08:49, 15 February 2006 (UTC)

Sounds fine, then. Go for it. Dysprosia 11:59, 15 February 2006 (UTC)

OK, I've moved the article to commutator subgroup. --Zundark 12:47, 15 February 2006 (UTC)

[edit] Notation

As pointed out above, the notation G¹ 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.

[edit] '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)

[edit] 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)