Talk:Circle group

Isomorphism equation and references

We should separate out the isomorphism

$\mathbb {T} \cong \mathbb {R} \oplus (\mathbb {Q} /\mathbb {Z} )\cong \mathbb {C} ^{\times }$

from the others as it is purely an algebraic isomorphism. As Lie groups T and C^× are certainly not isomorphic. This is bound to confuse readers. -- Fropuff 23:51, 30 November 2005 (UTC)

Does this article have any references at all? If not, it's certainly not a GA :/. Homestarmy 00:50, 12 May 2006 (UTC)

Okay, sorry for jumping the gun then. It's all common knowledge among mathematicians, but I'll go find a textbook to cite anyway. It'll give me a change to learn how to use this new-fangled <references/> thing. —Keenan Pepper 03:52, 12 May 2006 (UTC)

Is that formula correct at all? In other references [1] I find

$\mathbb {T} \cong \mathbb {(} R/\mathbb {Q} )\oplus (\mathbb {Q} /\mathbb {Z} )\cong \mathbb {C} ^{\times }$

Garrapito (talk) 15:59, 17 March 2014 (UTC)

Inconsistent angle units

It's a bit bizarre to have the first section measure angles in degrees, only to have later sections implicitly measure them in radians as is logical in this context.178.238.175.211 (talk) 09:23, 28 May 2016 (UTC)

I made radians explicit in the later section. Tom Ruen (talk) 11:52, 28 May 2016 (UTC)

Representations Section

The Representations section of this article makes it sound as if the irreducible representations of the circle group include both 1 and 2 dimensional representations. However, I thought---being an abelian group---that all of its irreps must be 1 dimensional. Am I missing something or is the article just worded in a confusing (possibly incorrect) way? Milez (talk) 19:41, 31 October 2013 (UTC)

Section: Representations

Somebody should tell us what the difference is between $\rho$, $\rho _{n}$, and $\phi _{n}$. Otherwise, we mix them all up. – Nomen4Omen (talk) 14:00, 16 August 2020 (UTC)

Elementary introduction

The first example of adding the angles is clear and easily understood. It could talk about modulo arithmetic when "adjusting" the sum to the range 0-360°.

The second example is confusing. Are we adding real or complex numbers? I would assume the numbers are in the set T but don't see the imaginary part. How do we get to the final result 0.155 from the previous 2.155? Throw away 2 does not explain anything. Jonne6v (talk) 07:15, 13 November 2022 (UTC)

I tried to make it clearer. –Nomen4Omen (talk) 10:18, 13 November 2022 (UTC)

One question I have is: When people say “circle group” do they necessarily mean the unit complex numbers per se? Or do they mean the group of orientations or rotations in the plane, for which the unit complex numbers are one possible representation (among many isomorphic alternatives)? This depends on context, and I am sure there are many sources which define it that way (in general, mathematicians are usually pretty cavalier about glossing the difference between isomorphic structures), but I guess my question would be: what would the most pedantic experts say is the proper definition / conception of the “circle group”? –jacobolus (t) 16:21, 13 November 2022 (UTC)