Talk:Orthogonal group

Relation (if any) to unitary groups
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)
 * 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)

Nothing about representations
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)


 * 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)

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

What are "hyperbolic lines" and "singular vectors"?
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:


 * $$V = L_1 \oplus L_2 \oplus \cdots \oplus L_m \oplus W,$$

where $L_{i}$ 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.


 * I hope someone knowledgeable on this topic will remedy these omissions.50.205.142.35 (talk) 14:37, 31 October 2019 (UTC)
 * I agree that this section is a mess, as well as a large part of the article. Here the confusion arise already from "If $V$ is the vector space on which the orthogonal group $G$ acts": If the orthogonal group is defined as the subgroup of $$GL(n,K)$$ consisting of the matrices such that $$^T\cdot T = I$$, then the orthogonal group acts on every vector space of dimension $n$. The structure of this group in the case of a finite field may be interesing, but it seems that this is not the subject of the section. The subgroup of $$GL(n,K)$$ under which a given qudratic form is invariant is often called the "orthogonal group" of the quadratic form. In this case, the assertion is probably a variant of Witt's decomposition theorem. A "hyperbolic line" is probably a vector line on which the quadrtic form is zero, that is an isotropic line, or a vector line that contains an isotropic vector (a hyperbolic plane is a plane that contains an isotropic vector). I guess that a "singular vector" is another name for an isotropic vector.
 * I will tag the section as confusing. D.Lazard (talk) 18:28, 31 October 2019 (UTC)

Rewrite of the article
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)

Section "Topology"
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)


 * 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)

Orthogonal groups over finite fields
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)


 * I have added a (collapsed) proof of the structure of $O±(2, q)$. D.Lazard (talk) 10:53, 20 November 2019 (UTC)
 * In fact, the order of $O(1, q)$ is two, and the formula given in the article is correct for this case.
 * The formulas given in the articles for even dimensions are correct for $n = 1$ if one replaces the case distinction "–1 is a square or not" by the distinction between $O+$ and $O–$. Therefore, I'll changing the case distinction and adapt the tags. D.Lazard (talk) 11:47, 21 November 2019 (UTC)
 * Not sure what the goal is, but the order formulae still have this source missing thing. You can cite the already used Taylor, The Geometry of Classical Groups, p. 141. The formula for $O-$ is off. Taylor gives 2q^2(q^2+1)(q^2-1), while the formula in the article gives 2(q^2-1)(q^4-q^2) = 2q^2(q^2-1)(q^2-1) which is clearly different. 141.134.35.237 (talk) 19:54, 15 April 2020 (UTC)
 * I have replaced the formulas in the article by Taylor's ones. For $O-$, I have shifted the index $n$ by one, because of the strange choice of Taylor of giving the formula for $O-(2m + 2, q)$. I hope not having done any error in copying and shifting the formulas. D.Lazard (talk) 13:40, 17 April 2020 (UTC)

Point inversion statement
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)

second paragraph in special orthogonal group
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)


 * Clarified (the notation $$\backslash$$ is standard, but not needed here). D.Lazard (talk) 11:46, 30 September 2022 (UTC)

third section of "special orthogonal group"
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)
 * Your impression is correct for $$n>2,$$ but wrong for $$n=2,$$ because $SO(2)$ 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)