Timeline for Reductive groups over positive characteristics
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Jun 7, 2022 at 16:57 | comment | added | Richard Lyons | For Q3, if |k|=3 then G(k) is still perfect unless G=A_1. See "Simple Groups of Lie Type" by the late Roger W. Carter. | |
| Jun 7, 2022 at 7:13 | comment | added | Peter McNamara | For Q3, if |k|>3 then G(k) is perfect under your hypotheses, for in this case G(k) is generated by its root SL_2(k)'s (by the Steinberg presentation), and each SL_2(k) is perfect. | |
| Jun 6, 2022 at 11:45 | answer | added | Peter McNamara | timeline score: 3 | |
| Jun 6, 2022 at 11:19 | history | edited | Dr. Evil | CC BY-SA 4.0 |
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| Jun 6, 2022 at 11:01 | history | edited | Dr. Evil | CC BY-SA 4.0 |
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| Jun 6, 2022 at 9:36 | comment | added | Matthieu Romagny | Ahah thank you Alexander and Peter for clarifying! Indeed I read too fast. | |
| Jun 6, 2022 at 8:08 | comment | added | Peter McNamara | 1 is not true as stated here in full generality, e.g. if |k|=2 and G is not a torus, or if |k|=3 and G is Sp_2n. Milne 21.1 is about the scheme-theoretic Weyl group and in this question we're talking about normalisers within the group of k-points. My guess is that 1 is true if |k|>3. | |
| Jun 6, 2022 at 8:07 | comment | added | A Stasinski | @Matthieu Romagny I don't think that's true, since if $k$ has two elements and $T$ is the diagonal torus in $G=\mathrm{GL}_2$, we have $T(k)=\{1\}$ so $N_{G(k)}(T(k))=G(k)$. Note that Milne is talking about $N_G(T)(k)$, which is not always isomorphic to $N_{G(k)}(T(k))$. | |
| Jun 6, 2022 at 6:59 | comment | added | Matthieu Romagny | Assertion 1. is true with no condition on $k$, see Milne's book 'Algebraic groups' (CUP), Prop. 21.1. | |
| Jun 6, 2022 at 6:09 | history | asked | Dr. Evil | CC BY-SA 4.0 |