Talk:Normal subgroup
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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
- 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)
[edit] Theorem statement
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)
[edit] "Normality" property
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.
--—The preceding unsigned comment was 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. --Chan-Ho (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)
