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The following sentence from the article is wrong; all of the conditions are equivalent, so one can't be weaker than another. (I can't think of a better way to say it, though, so I'm leaving it as is for now.) Cwitty
Regarding this, I think the current article is correct but I don't think it is so relevant. It may be helpful to be aware of those conditions in writing a proof but is irrelevant to the general discussion of a normal subgroup, I think. It's even distracting a bit. If no one opposes, I am going to trim the list. -- Taku 23:25, 4 October 2005 (UTC)
Now I think the difference lies in distinguishing “if we assume the group axioms and normality of N” versus “if we make weaker assumptions”. Formally: let , … denote theories, , … denote statements. Let us define also the followings notations
Now let denote the set of group axioms. and let be defined as group axioms plus normality of N:
Let be defined as a “sufficiently weak” theory. E.g. empty
or the semigroup axioms
or something like such, I am not sure now the exact calibration of strength.
The debated, but really very important statements in the article want to say, I think, that
but
where can be substituted for ,
But this is only a momentary impression of mine, I am not sure yet.
In summary, I think the remark of the article about "different strength of finally equivalent statements" is very important. It helps the understanding of a new concept, when it is illustrated and approached at as many sides as possible, and the full machinery of the logical network of these approaches is revealed: interdependence of small lemma chips: what uses what, what assumes what, what is true inherently and what is true only by using a consequence of another fact etc.
Physis 01:47, 28 September 2007 (UTC)
Regarding the statement of equivalent conditions in the Defintions section: I do not believe that "N is a union of conjugacy classes of G" is equivalent to "N is a normal subgroup of G". The former may be implied by the latter, but e.g. any single nontrivial conjugacy class of a simple group satisfies the former but not the latter (unless I'm mistaken). — Preceding unsigned comment added by 24.218.56.194 ( talk) 21:30, 26 June 2012 (UTC)
From the article:
Is it possible to change the words "contains a subgroup K normal in G" to "contains a non-trivial subgroup K normal in G"? Or isn't that true? -- Quuxplusone 19:14, 10 November 2005 (UTC)
This paragraph is very confusing. How is a "normal subgroup" not "normal"? Either this is a mistake (made three times) and someone should fix it or it isn't and someone should define "normal".
A normal subgroup of a normal subgroup need not be normal. That is, normality is not a transitive relation. However, a characteristic subgroup of a normal subgroup is normal. Also, a normal subgroup of a central factor is normal. In particular, a normal subgroup of a direct factor is normal.
--—Preceding unsigned comment added by Varuna ( talk • contribs)
Read the first sentence as: A normal subgroup K of a normal subgroup H of G need not be normal in G. -- C S (Talk) 02:07, 4 August 2006 (UTC)
Thank you, Chan-Ho. If I find the time to do it right, I'll try to make the article clearer. -- Varuna 01:21, 7 August 2006 (UTC)
Hi,
I have recently "rediscovered" normal subgroups and simple groups and was rather disappointed to see that not much mathematical attention seems to be paid to the denoting and classification of groups which cannot be expressed as (or, in other words, are not isomorphic to) the direct product of any two or more smaller, nontrivial groups. I have posted a question about any special name for such groups in Talk:Simple group, but I thought I would ask here if there is any special name for a normal subgroup K (to distinguish it from N, and since K is the first letter of my first name) of a group G such that the direct product (rather than merely a semidirect product) of K and the quotient group G/K is isomorphic to G itself.
Groups containing no such subgroups besides the trivial group and the group itself, and only such groups, are not isomorphic to the direct product of any smaller, nontrivial groups. In my opinion such groups and such subgroups K (you could subsitute another letter) deserve a special name, if they don't have one already. They aren't characteristic subgroups, as any subgroup of the Klein four-group would qualify as such a subgroup (with the trivial group and the Klein four-group itself being each other's quotient groups within the group) even though each of the Klein four-groups three subgroups of order two are not characteristic. And thus they can't be distinguishing or fully characteristic subgroups. If anyone can help answer any of these questions I would appreciate it. Kevin Lamoreau 00:36, 21 January 2007 (UTC)
![]() | This page is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
The following sentence from the article is wrong; all of the conditions are equivalent, so one can't be weaker than another. (I can't think of a better way to say it, though, so I'm leaving it as is for now.) Cwitty
Regarding this, I think the current article is correct but I don't think it is so relevant. It may be helpful to be aware of those conditions in writing a proof but is irrelevant to the general discussion of a normal subgroup, I think. It's even distracting a bit. If no one opposes, I am going to trim the list. -- Taku 23:25, 4 October 2005 (UTC)
Now I think the difference lies in distinguishing “if we assume the group axioms and normality of N” versus “if we make weaker assumptions”. Formally: let , … denote theories, , … denote statements. Let us define also the followings notations
Now let denote the set of group axioms. and let be defined as group axioms plus normality of N:
Let be defined as a “sufficiently weak” theory. E.g. empty
or the semigroup axioms
or something like such, I am not sure now the exact calibration of strength.
The debated, but really very important statements in the article want to say, I think, that
but
where can be substituted for ,
But this is only a momentary impression of mine, I am not sure yet.
In summary, I think the remark of the article about "different strength of finally equivalent statements" is very important. It helps the understanding of a new concept, when it is illustrated and approached at as many sides as possible, and the full machinery of the logical network of these approaches is revealed: interdependence of small lemma chips: what uses what, what assumes what, what is true inherently and what is true only by using a consequence of another fact etc.
Physis 01:47, 28 September 2007 (UTC)
Regarding the statement of equivalent conditions in the Defintions section: I do not believe that "N is a union of conjugacy classes of G" is equivalent to "N is a normal subgroup of G". The former may be implied by the latter, but e.g. any single nontrivial conjugacy class of a simple group satisfies the former but not the latter (unless I'm mistaken). — Preceding unsigned comment added by 24.218.56.194 ( talk) 21:30, 26 June 2012 (UTC)
From the article:
Is it possible to change the words "contains a subgroup K normal in G" to "contains a non-trivial subgroup K normal in G"? Or isn't that true? -- Quuxplusone 19:14, 10 November 2005 (UTC)
This paragraph is very confusing. How is a "normal subgroup" not "normal"? Either this is a mistake (made three times) and someone should fix it or it isn't and someone should define "normal".
A normal subgroup of a normal subgroup need not be normal. That is, normality is not a transitive relation. However, a characteristic subgroup of a normal subgroup is normal. Also, a normal subgroup of a central factor is normal. In particular, a normal subgroup of a direct factor is normal.
--—Preceding unsigned comment added by Varuna ( talk • contribs)
Read the first sentence as: A normal subgroup K of a normal subgroup H of G need not be normal in G. -- C S (Talk) 02:07, 4 August 2006 (UTC)
Thank you, Chan-Ho. If I find the time to do it right, I'll try to make the article clearer. -- Varuna 01:21, 7 August 2006 (UTC)
Hi,
I have recently "rediscovered" normal subgroups and simple groups and was rather disappointed to see that not much mathematical attention seems to be paid to the denoting and classification of groups which cannot be expressed as (or, in other words, are not isomorphic to) the direct product of any two or more smaller, nontrivial groups. I have posted a question about any special name for such groups in Talk:Simple group, but I thought I would ask here if there is any special name for a normal subgroup K (to distinguish it from N, and since K is the first letter of my first name) of a group G such that the direct product (rather than merely a semidirect product) of K and the quotient group G/K is isomorphic to G itself.
Groups containing no such subgroups besides the trivial group and the group itself, and only such groups, are not isomorphic to the direct product of any smaller, nontrivial groups. In my opinion such groups and such subgroups K (you could subsitute another letter) deserve a special name, if they don't have one already. They aren't characteristic subgroups, as any subgroup of the Klein four-group would qualify as such a subgroup (with the trivial group and the Klein four-group itself being each other's quotient groups within the group) even though each of the Klein four-groups three subgroups of order two are not characteristic. And thus they can't be distinguishing or fully characteristic subgroups. If anyone can help answer any of these questions I would appreciate it. Kevin Lamoreau 00:36, 21 January 2007 (UTC)