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As pointed out above, the notation is unusual. If there are no objections, I intend to change it to (because the alternative is too easily misread as in some fonts). -- Zundark 12:49, 15 February 2006 (UTC)
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 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 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 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 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)
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)
This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
This page has archives. Sections older than 365 days may be automatically archived by Lowercase sigmabot III when more than 5 sections are present. |
As pointed out above, the notation is unusual. If there are no objections, I intend to change it to (because the alternative is too easily misread as in some fonts). -- Zundark 12:49, 15 February 2006 (UTC)
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 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 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 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 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)
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)