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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)
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)
Talk:Orthogonal group/Rotation group (disambiguation) per Wikipedia:Articles for deletion/Rotation group (disambiguation). Sandstein 18:47, 25 October 2018 (UTC)
The section Over finite fields contains this sentence:
"If V is the vector space on which the orthogonal group G acts, it can be written as a direct orthogonal sum as follows:
where Li are hyperbolic lines and W contains no singular vectors."
But the article never defines either "hyperbolic lines" or "singular vectors".
For that matter, the article never defines an orthogonal group over a finite field. We are told some things about such groups, but we are never told how they are defined.
Also: I just noticed that the article treats KO as a topological space (describing the homotopy groups πk(KO), but never defines what KO is, either.
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)
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, SO(2) is sometimes denoted S1, T1, U1, 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)
I have rewritten the part of the section devoted to characteristic different from two in a way that allows verifying the results without searching in the references. Several results remain unclear for me.
The structure of O±(2, q): I unable to provide an isomorphism with the dihedral group. It seems that the number of elements relies on the number of points of a non-degenerate conic (that is q + 1), and on the number of regular points of a pair of intersecting lines (that is 2(q – 1)), but I am unable to explain this sufficiently brievly for this article.
The order of the orthogonal groups: The given formulas are dubious: There is no distinction between O+(2n, q) and O–(2n, q), although, in the preceding paragraph, it is said that they have not the same order for n = 1. Also the formula for O(2n + 1, q) gives 2 instead of q – 1 for n = 0.
D.Lazard ( talk) 16:30, 14 November 2019 (UTC)
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)
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)
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)
thanks! understood. une musque de Biscaye ( talk) 16:32, 29 September 2022 (UTC)