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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeApr 15th 2018
• (edited Apr 15th 2018)

I am splitting off an entry classification of finite rotation groups from ADE classification in order to collect statements and references specific to the classification of finite subgroups of $SO(3)$ and $SU(2)$.

Is there a canonical reference for the proof of the classification statement? I find lots of lecture notes that give the proof, but all of them without citing sources or original publications of proofs.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeApr 15th 2018

found a good textbook account: Rees 05

but again there is no pointer in there to the origin of the statement.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeApr 15th 2018

I wrote:

Is there a canonical reference for the proof of the classification statement? I find lots of lecture notes that give the proof, but all of them without citing sources or original publications of proofs.

In Burban’s lecture notes (pdf) it has this statement:

The classification of finite isometry groups of $\mathbb{R}^3$ is a classical result of F. Klein (actually of Platon)

but still no concrete pointer…

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeApr 15th 2018

according to Lamotke’s book, it seems that the result goes back to Klein 1884

(have not checked yet, need to run now)

• CommentRowNumber5.
• CommentAuthorDavid_Corfield
• CommentTimeApr 15th 2018

From here

In his Vorlesungen über das Ikosaeder [Klein,1993], published in 1884, Felix Klein gives the classification of finite subgroups of $SL(2,\mathbb{C})$ up to conjugacy

There’s an online translation here.

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeApr 15th 2018

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeApr 16th 2018

Thanks for this! Excellent. I have added that reference also to all the entries on the separate finite rotation groups, as well as to that of the corresponding platonic solids.

• CommentRowNumber8.
• CommentAuthorDavidRoberts
• CommentTimeApr 16th 2018

Surely the binary cyclic groups (ie the even-order cyclic groups) are also pre-images of the cyclic group of half the order?

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeApr 17th 2018

Yes, but not every cyclic subgroup of $SU(2)$ is in the image of taking pre-images. The odd order cycile groups are clearly not.

• CommentRowNumber10.
• CommentAuthorDavidRoberts
• CommentTimeApr 17th 2018

Tweaked description of which finite subgroups are preimages under the double cover projection.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeApr 17th 2018

Oh, now I see. Thanks!

• CommentRowNumber12.
• CommentAuthorDavidRoberts
• CommentTimeApr 17th 2018

For what it’s worth I think the odd-order cyclic subgroups of $SU(2)$ are the image of the section $\mathbb{Z}/(2k+1) \to \mathbb{Z}/(4k+2) \simeq \mathbb{Z}/(2k+1) \times \mathbb{Z}/2$, so the odd-order cyclic subgroups of $SU(2)$ are index-2 subgroups of inverse images of finite subgroups of $SO(3)$, and the projection map $SU(2) \to SO(3)$ restricts to an isomorphism on these odd-order cyclic subgroups.