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nLab > Latest Changes: classification of finite rotation groups

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  1.  
    • 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)SO(3) and SU(2)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.

    v1, current

    • 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.

    diff, v2, current

    • 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 ℝ 3\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)

    diff, v3, current

    • 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,ℂ)SL(2,\mathbb{C}) up to conjugacy

    There’s an online translation here.

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 15th 2018

    I added bibliographic detail about Klein’s work and the translation.

    diff, v4, current

    • 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)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.

    diff, v9, current

    • 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)SU(2) are the image of the section ℤ/(2k+1)→ℤ/(4k+2)≃ℤ/(2k+1)×ℤ/2\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)SU(2) are index-2 subgroups of inverse images of finite subgroups of SO(3)SO(3), and the projection map SU(2)→SO(3)SU(2) \to SO(3) restricts to an isomorphism on these odd-order cyclic subgroups.

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