Timeline for Classification of finite groups of isometries
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| S May 27, 2016 at 16:32 | history | suggested | Sean Lawton | CC BY-SA 3.0 |
Minor edits: fixed uncompiled TeX, formatting, added tag.
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| May 27, 2016 at 16:13 | review | Suggested edits | |||
| S May 27, 2016 at 16:32 | |||||
| Jan 20, 2013 at 19:55 | answer | added | Günter Rote | timeline score: 7 | |
| Apr 23, 2011 at 11:41 | answer | added | Geoff Robinson | timeline score: 14 | |
| Aug 30, 2010 at 19:38 | comment | added | Vaughn Climenhaga | It's also worth pointing out that if G is any finite subgroup of GL(n,R), then one can introduce an inner product with respect to which G acts by isometries; in particular, there is an inner automorphism of GL(n,R) that maps G to a finite subgroup of Isom(n,R). Thus classifying finite groups of isometries is equivalent to classifying finite groups of linear transformations. | |
| Aug 30, 2010 at 19:35 | comment | added | Vaughn Climenhaga | @unknown: If $G \subset GL(n,\mathbb{R})$ is finite, then we may change coordinates so that the origin is at the centre of mass of some orbit $Gx$ (for an arbitrary $x\in \mathbb{R}^n$). Then the origin is fixed by every element of $G$, so linearity follows from finiteness. On the other hand, if you want to consider discrete subgroups, then you do need to allow for the possibility that some of your isometries may be affine transformations. | |
| Aug 30, 2010 at 17:10 | history | edited | Charles Matthews | CC BY-SA 2.5 |
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| Aug 30, 2010 at 16:19 | answer | added | Richard Borcherds | timeline score: 45 | |
| Aug 30, 2010 at 15:35 | comment | added | Will Orrick | Scratch my earlier comment about Platonic solids. I was thinking of finite rotational symmetries only. Including reflections obviously allows richer structures. In three dimensions, for example, you get finite versions of the frieze groups, among other things. (See en.wikipedia.org/wiki/…) Does this really go all the way back to Kepler? I suspect it's much more recent than that. | |
| Aug 30, 2010 at 13:51 | comment | added | Jim Humphreys | As Jose points out, the current question is largely covered by a previous one; many references are given there. It's best to formulate your own question more precisely as a follow-up if needed. | |
| Aug 30, 2010 at 12:51 | comment | added | Will Orrick | John Baez has a page on the analogues of the Platonic solids in all dimensions: math.ucr.edu/home/baez/platonic.html. The pattern becomes regular starting in dimension 5. I haven't looked at it, but Coxeter's book H. S. M. Coxeter, Regular Polytopes, 3rd edition, New York, Dover Publications, 1973, probably contains more information. | |
| Aug 30, 2010 at 12:46 | comment | added | José Figueroa-O'Farrill | You may also look at the references in this answer to a previous MO question: mathoverflow.net/questions/17072/the-finite-subgroups-of-sun/… | |
| Aug 30, 2010 at 12:39 | answer | added | Roland Bacher | timeline score: 9 | |
| Aug 30, 2010 at 11:29 | comment | added | José Figueroa-O'Farrill | The finite subgroups of SO(4) are listed in the book by Conway and Smith on quaternions and octonions. I have recently checked that it is correct and recovered it from the classification of finite subgroups of Spin(4), which we needed for a separate project: see the paper arxiv.org/abs/1007.4761 . | |
| Aug 30, 2010 at 9:33 | history | asked | Mathieu Dutour Sikiric | CC BY-SA 2.5 |