Timeline for Sylow subgroups of orthogonal group
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Dec 1, 2019 at 15:40 | comment | added | LSpice | A sketch proof: $G/B$ is paved by affines (Bruhat decomposition), one of which is 0-dimensional, so has prime-to-$p$ order, so $B$ contains a $p$-Sylow. $B/U \cong T$ consists of semisimple elements, so has prime-to-$p$ order. | |
| Nov 11, 2019 at 19:27 | comment | added | kneidell | @TheoJohnson-Freyd is correct. See, for example, Corollary 24.11 in Linear algebraic groups and finite groups of Lie-type, by Testerman and Malle, and use the fact that your group acts transitively by conjugation on its $p$-Sylow subgroups and on radical unipotents. | |
| Nov 11, 2019 at 14:28 | comment | added | Richard Lyons | Your wiki reference doesn't do justice to the finite orthogonal groups. R Carter's "Simple Groups of Lie Type" and D Taylor's "The Geometry of the Classical Groups" are two fine books that answer your questions. E.g. in even dimensions $n$, $O^+(n,q)$ and $O^-(n,q)$ are different groups, with different orders & different Sylow structures. Given a vector space $V$ and a nondegenerate quadratic form $Q$ on $V$, the orthogonal group $O(V,Q)$ is the subgroup of $GL(V)$ preserving $Q$. If $dim(V)$ is even there are two inequivalent quadratic forms on $V$. The wiki page only considers one of them. | |
| Nov 11, 2019 at 5:44 | comment | added | Theo Johnson-Freyd | An expert may correct me, but I think p-Sylow in a group defined in characteristic p is usually the unipotent subgroup (upper triangular matrices). | |
| Nov 10, 2019 at 19:34 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
copied the relevant part of the WP article
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| Nov 10, 2019 at 19:28 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
added the Wikipedia link
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| Nov 10, 2019 at 19:22 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
Removed the deprecated (abstract-algebra) tag - see the tag info: https://mathoverflow.net/tags/abstract-algebra/info (if there are some other suitable tags, choose them instead.)
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| Nov 10, 2019 at 18:40 | review | First posts | |||
| Nov 10, 2019 at 19:42 | |||||
| Nov 10, 2019 at 18:36 | history | asked | Zachary | CC BY-SA 4.0 |