Timeline for What finite simple groups we can obtain using octonions?
Current License: CC BY-SA 4.0
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| Apr 9, 2019 at 7:47 | history | edited | user21230 | CC BY-SA 4.0 |
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| Apr 9, 2019 at 7:42 | history | edited | user21230 | CC BY-SA 4.0 |
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| Jan 3, 2019 at 9:38 | comment | added | user21230 | I am afraid of editing my answer - it seems editing is persecuted on this forum - so I add new information as a comment. The number of norm 1 and trace -1 octonions is $q^6-q^3, q^6+q^3, q^6-1$ and stabilizer is $3.L_3(q), 3.U_3(q)$ or other for $q=1,2,0(mod 3)$. It would be interesting to connect maximal subgroups of $G_2(q)$ to subgroups fixing some subalgebras of octonions. What objects are fixed by sporadics $J_2$ and $J_1$ embedded in $G_2(4)$ and $G_2(11)$ ? | |
| Jan 2, 2019 at 21:35 | history | edited | user21230 | CC BY-SA 4.0 |
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| Jan 2, 2019 at 15:02 | comment | added | user21230 | Over finite field there is just one version of octonions as far as I remember. In odd characteristic you can use both definitions but you receive the same algebra anyway. In characteristic two there is just one definition you should use - the one for split octonions - which you can extend to characteristic 2. I am not sure what similar algebras of dimension 8 exist over each of the finite field. I have not done any systematic research here. What I know for sure is that split octonions definition can be used over any field. See function OctaveAlgebra in GAP which defines such algebra. | |
| Jan 2, 2019 at 14:23 | comment | added | მამუკა ჯიბლაძე | So there is no analog of the fact that over $\mathbb R$ there are split and non-split octonions which only become isomorphic over $\mathbb C$? | |
| Jan 2, 2019 at 13:34 | history | edited | user21230 | CC BY-SA 4.0 |
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| Jan 2, 2019 at 13:15 | history | edited | user21230 | CC BY-SA 4.0 |
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| Jan 2, 2019 at 13:01 | history | edited | user21230 | CC BY-SA 4.0 |
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| Jan 2, 2019 at 12:59 | comment | added | user21230 | Yes, in field GF(5) we have $2*2=-1$ and it works as well. | |
| Jan 2, 2019 at 12:50 | comment | added | მამუკა ჯიბლაძე | Does this work regardless of whether the field contains square roots of $-1$ or not? | |
| S Jan 2, 2019 at 12:15 | history | answered | user21230 | CC BY-SA 4.0 | |
| S Jan 2, 2019 at 12:15 | history | made wiki | Post Made Community Wiki by user21230 |