Timeline for Poincaré dodecahedron space
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Mar 22, 2012 at 1:26 | comment | added | Alex Suciu | Or, one could cite <en.wikipedia.org/wiki/Binary_icosahedral_group> or <groupprops.subwiki.org/wiki/Special_linear_group:SL(2,5)> (if url's would work nicely in this markup language, that is...) | |
| Mar 22, 2012 at 1:07 | comment | added | Alex Suciu | Well, I have to confess, I found this isomorphism with the help of GAP, which assures me this map (which can be checked "by hand" is a homomorphism) is indeed a bijection. I would need more motivation to verify this by hand, but surely it can be done. Now, as to why $I^*$ is isomorphic to ${\rm SL}(2,5)$. One could use the same kind argument: start with a known presentation of $I^*$, say, $\langle x, y \mid x^3=y^5=(xy)^2 \rangle$, write down a map to ${\rm SL}(2,5)$, and check it's an isomorphism. Or, one could use a more geometric approach (symmetries of the icosahedron, quaternions, etc). | |
| Mar 21, 2012 at 22:39 | comment | added | Paul | @ Alex: What is the simplest way to see that the map you wrote is injective? You are right that the fact that this presents $I^*$ is "well-known", but the arguments I know involve some geometry, eg Goodwillie's comment. | |
| Mar 21, 2012 at 5:22 | history | edited | Alex Suciu | CC BY-SA 3.0 |
added 4 characters in body
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| Mar 21, 2012 at 5:17 | history | answered | Alex Suciu | CC BY-SA 3.0 |