Talk:Orthogonal group

message: Given that unitary matrices preserve the standard inner product on complex matrices, it seems plausible to suggest that the complex orthogonal groups are related to the unitary groups.  If this is in any sense the case, should it not be qualified?--Leon (talk) 13:12, 13 September 2010 (UTC)
replies: text: Yes, they are, in a large and rich number of ways. Perhaps the article gets into that a bit, now. 67.198.37.16 (talk) 18:38, 2 September 2016 (UTC)
thread_title:  Relation (if any) to unitary groups 

message: Nothing is said about the representations of orthogonal group
I would write sth if I knew enough
Massive Fermion (talk) 06:32, 7 September 2012 (UTC)
replies: text: Oof-dah. Maybe classical groups talks about this.  Is there a representations of classical groups or representation theory of classical groups or representations of Lie groups or representation theory of Lie groups? Only the blue links know... 67.198.37.16 (talk) 18:44, 2 September 2016 (UTC)
thread_title:  Nothing about representations 

message: Talk:Orthogonal group/Rotation group (disambiguation) per Wikipedia:Articles for deletion/Rotation group (disambiguation).  Sandstein  18:47, 25 October 2018 (UTC)
thread_title:  Talk page moved here 

message: Because of the problem pointed in the preceding thread, I have edited a large part of the article. The objective was to be able to understand the article myself without using my (rather poor) mathematical knowledge of the subject. By "understanding", I mean being able to verify the results with the indications given in the article and the linked articles. I hope to have succeeded in the sections that I have edited.
I have not edited the sections from the one about Dickson invariant on, and, at least for the moment, I leave this work to other editors. Two of the first sections remain problematic, at least partially. I'll discuss their issues in separate threads
I hope that other editors will improve and continue my work, here and in the linked articles. In fact, I have  added some content that belong normally to linked articles. Sometimes, it is because repeating them here can make reading easier, but, in some cases, this is because I have not found, in Wikipedia, a correct presentation of the needed background. This is the case, for example for the classification of the quadratic forms over a finite field. D.Lazard (talk) 15:33, 14 November 2019 (UTC)
thread_title:  Rewrite of the article 

message: The present state of the section can be summarized as follow: Orthogonal groups are topological spaces; so all algebraic topology applies; so, one lists all groups that can be defined from othogonal groups in algebraic topology, without any organization nor indication of the relevance for the study of orthogonal groups. Moreover, most notations are not defined nor linked, and they are not really harmonized. For example,  is sometimes denoted , , , or called the circle group. The result is boring and not useful for a reader that is not a specialist of the subject.
So, a complete rewrite of this section is needed. I am unable to do this myself. D.Lazard (talk) 15:54, 14 November 2019 (UTC)
replies: text: It's not at all boring. It's a handy-dandy reference/cheat-sheet if you're doing physics, and want to double check something to make sure all of your calculations are coming out as expected. It would be easy if everything worked exactly the same way in all dimensions. But it doesn't. 67.198.37.16 (talk) 05:47, 15 November 2023 (UTC)
thread_title:  Section "Topology" 

message: The statement about reflection through the origin "The reflection through the origin (the map v ↦ −v) is an example of an element of O(n) that is not the product of less than n reflections." is very convoluted wording. Can it be simplified whilst still remaining correct? UphillPhil (talk) 10:31, 24 June 2020 (UTC)
thread_title:  Point inversion statement 

message: maybe the section
Moreover, the orthogonal group is a semidirect product of SO(n) and the group with two elements, since, given any reflection r, one has O(n) \ SO(n) = r SO(n).
could be extended and explained better. what is O(n) \ SO(n)? what does r SO(n) mean and what is the relationship with the semidirect product? If r SO(n) is a coset, then the quotient group is the group of all quotients, not just a coset (there should be two cosets I think). If rSO(n) is the product group, I think this is not the quotient O(n)/SO(n).une musque de Biscaye (talk)22:22, 27 September 2022 (UTC)
replies: text: Clarified (the notation \backslash is standard, but not needed here). D.Lazard (talk) 11:46, 30 September 2022 (UTC)
thread_title:  second paragraph in special orthogonal group 

message: For every positive integer k the cyclic group Ck of k-fold rotations is a normal subgroup of O(2) and SO(2).
Can this be expanded a bit? I was under the impression that the only normal subgroups of O(n) are SO(n), {I} and {I, -}
https://math.stackexchange.com/questions/2105981/normal-subgroups-of-on une musque de Biscaye (talk) 22:26, 27 September 2022 (UTC)
replies: text: Your impression is correct for n>2, but wrong for n=2, because  is commutative. I agree that the formulation was confusing, and I have edited the article for clarifying it.
thanks! understood. une musque de Biscaye (talk) 16:32, 29 September 2022 (UTC)
thread_title:  third section of "special orthogonal group"