|WikiProject Mathematics||(Rated C-class, High-importance)|
Formalization of informal treatment of “stength of equivalent statements”
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
- Note that condition (1) is logically weaker than condition (2), and condition (3) is logically weaker than condition (4).
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)
- Let us keep them and illustrate them with examples.--Patrick 23:58, 4 October 2005 (UTC)
- The least I can say is I will try to find time to do this :) Others are certainly welcome as well. -- Taku 07:44, 6 October 2005 (UTC)
- Just wanted to say that the condirions were exactly what I wanted from the article when trying to decode my lecture notes on the subject. Kvetch 14:56, 5 January 2007 (UTC)
- I agree the list of conditions is helpful, but the claim that the conditions are somehow logically equivalent and logically weaker at the same time is confusing. I now do understand why the article is correct. To use meta logical language, the claims 1 and 2 have the same extension, but they do not have the same intension. This is really quite settled, and is really only relevant in relation to the mentioned proof techniques. 184.108.40.206 11:35, 14 August 2007 (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
- Let denote
- Let denote
- Let denote
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
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 220.127.116.11 (talk) 21:30, 26 June 2012 (UTC)
From the article:
- Every subgroup of index 2 is normal. More generally, a subgroup H of finite index n in G contains a subgroup K normal in G and of index dividing n!.
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)
- It's obviously not true in general, since H itself could be trivial. --Zundark 08:40, 11 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.
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)
Special name for a normal subgroup K of a group G whose DIRECT product with the quotient group G/K is isomorphic to G itself?
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)
- I remember asking my graduate algebra teacher the question: "For normal subgroup N, is there a subgroup of G that looks like G/N?" And the answer is, of course, not always. If I remember right, I think the prof used the word "decomposable" -- check out Direct sum of groups. Maybe that will help. - grubber 02:37, 21 January 2007 (UTC)
- That helped a little, thanks. Someone else answered my other question in Talk:Simple group, in case anyone was interested. Such groups are indeed called indecomposable groups (you had said decomposable, referring to groups which are isomporphic to the direct product of smaller, nontrivial subgroups, but were definitely on the money). Kevin Lamoreau 16:28, 21 January 2007 (UTC)
- I think the other term you are looking for is internal direct factor, although I don't think your definition is precisely this. I presume you are really thinking of the case in which K x G/K is isomorphic to G via a natural canonical map which takes each element of the form (k,1) to k. I suspect, though, that it is also possible for K x G/K to be coincidentally isomorphic to G in a completely different way. (So the abstract group K would be a direct factor, but the particular subgroup K would not be an internal direct factor - of course G would have another subgroup isomorphic to K which was a true direct factor.) I guess an alternative definition, which avoids these tricky isomorphism issues, is to say that K is an internal direct factor if it is one of a pair of subgroups which commute, intersect trivially and together generate G. (The other will of course be isomorphic to G/K.) Best wishes, Cambyses 20:22, 22 January 2007 (UTC)
- Thanks, Cambyses. I'm not at your level mathematically I'm afraid, at least not with groups. I actually have a B.A. in Mathematics, but my grades weren't the best (I had little self-discipline regarding completing assignments or studying) and groups weren't my strongest suit, although they definitely intrigued me and I have recently studied certain aspects of group theory out of personal interest. I happened to be more interested in what groups that weren't the direct products of any smaller nontrivial groups were called (indecomposable groups) and asked the question here to get more information towards answering that question. Thank you though for the information you did provide. I may eventually learn enough of the lingo (my interest is right now more about group classification) to get everything you wrote. Kevin Lamoreau 04:23, 24 January 2007 (UTC)