Timeline for Does the map $\mathrm{SL}(n,\mathbb{Z}/p^2) \rightarrow \mathrm{SL}(n,\mathbb{Z}/p)$ split?
Current License: CC BY-SA 4.0
24 events
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| Mar 29, 2023 at 15:14 | vote | accept | BasicQuestionBot | ||
| Mar 29, 2023 at 8:40 | answer | added | A Stasinski | timeline score: 10 | |
| Mar 28, 2023 at 18:00 | history | edited | BasicQuestionBot | CC BY-SA 4.0 |
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| Mar 28, 2023 at 17:44 | answer | added | ahulpke | timeline score: 8 | |
| Mar 28, 2023 at 15:00 | comment | added | BasicQuestionBot | @DavidESpeyer: That's a really nice way to think about it, thanks! | |
| Mar 28, 2023 at 15:00 | comment | added | BasicQuestionBot | @YCor: Ah, that's a cool theorem I didn't know about! I guess it follows from the fact that the the $p$-primary part of group cohomology is supported on a $p$-Sylow subgroup. | |
| Mar 28, 2023 at 13:55 | comment | added | David E Speyer | Regarding: "I don't have any deep explanation as to why this works". $\text{SL}_2(\mathbb{Z}/2 \mathbb{Z})$ is isomorphic to $S_3$, which has a $2$-dimensional representation over the integers. Unfortunately, that representation involves matrices with determinant $1$ . | |
| Mar 28, 2023 at 13:34 | answer | added | David E Speyer | timeline score: 6 | |
| Mar 28, 2023 at 12:05 | answer | added | YCor | timeline score: 10 | |
| Mar 28, 2023 at 11:11 | comment | added | YCor | Ah, OK, in view of the given observations, for $n=2$ it only splits for $p=3$ (Gaschütz applies precisely iff the elementary matrix $e_{12}(1)$ lifts). | |
| Mar 28, 2023 at 8:20 | answer | added | Dave Benson | timeline score: 17 | |
| Mar 28, 2023 at 7:07 | comment | added | YCor | @BasicQuestionBot Essentially copying as in math.ku.dk/~olsson/manus/Gruppe-2009/gruppe2007en_all.pdf, p16: let $p$ be a prime and let $M$ be a normal abelian $p$-subgroup of a finite group $G$. Then $M$ has a direct complement in $G$ if (and only if) it has a direct complement in some/any $p$-Sylow subgroup of $G$. | |
| Mar 28, 2023 at 1:14 | comment | added | BasicQuestionBot | @YCor: Interesting, which theorem of Gaschütz are you thinking of? | |
| Mar 28, 2023 at 1:13 | comment | added | BasicQuestionBot | @DavidESpeyer: Whoops, you're right! I'll have to go back and see if I can fix this. | |
| Mar 28, 2023 at 1:09 | comment | added | David E Speyer | The splitting you give for $p=2$ give matrices with determinant $-1$. | |
| Mar 28, 2023 at 1:09 | comment | added | YCor | @LSpice yes it splits for $(n,p)=(2,2),(2,3)$: by Gaschütz' theorem it's enough to do it on $p$-Sylows of the quotient, which are cyclic of order $p$, and indeed split. | |
| Mar 28, 2023 at 1:02 | history | edited | LSpice | CC BY-SA 4.0 |
`SL` -> `\SL`
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| Mar 28, 2023 at 0:51 | comment | added | BasicQuestionBot | @LSpice: I added the answer I know there to the question. | |
| Mar 28, 2023 at 0:50 | history | edited | BasicQuestionBot | CC BY-SA 4.0 |
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| Mar 27, 2023 at 23:03 | comment | added | LSpice | Do you know the answer for $n = 2$? | |
| Mar 27, 2023 at 22:06 | review | Close votes | |||
| Apr 1, 2023 at 3:11 | |||||
| Mar 27, 2023 at 21:41 | history | edited | YCor | CC BY-SA 4.0 |
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| S Mar 27, 2023 at 21:14 | review | First questions | |||
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| S Mar 27, 2023 at 21:14 | history | asked | BasicQuestionBot | CC BY-SA 4.0 |