Timeline for Finding an overgroup or a subgroup in PGL
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| Nov 26, 2014 at 16:26 | comment | added | Nick Gill | Tomo, I think $H'$ does have an infinite number of overgroups - you just need to caclulate its normalizer to see this... But I haven't time to do the details... If you restrict to connected overgroups then the answer might be different... | |
| Nov 25, 2014 at 19:08 | comment | added | Tomo | Thanks very much for the comment. I have another question. Now let $H'$ be the one dimensional unipotent subgroup of $H$ obtained by taking $y=w=0$. Are there infinitely many overgroups of $H'$ in $PGL_4(k)$? Please forget my last comment on finding overgroups of $H$ in $PGL_4(\bar {k})$. | |
| Nov 25, 2014 at 1:04 | comment | added | Nick Gill | For an overgroup take the normalizer of the connected subgroup of dimension $1$ that I describe above. | |
| Nov 24, 2014 at 17:07 | comment | added | Nick Gill | It might be worth mentioning in the question that $H$ is abelian in general. | |
| Nov 24, 2014 at 17:05 | comment | added | Nick Gill | OK, for a connected subgroup of dimension 1 just take $y=w=0$. The resulting set is is an abelian unipotent subgroup (in particular it has exponent $2$). In fact, if I am not mistaken, over $\overline{k}$ it looks like a root group for $PSp_4(k)$. (Although the question over $\overline{k}$ needs clarification because in this context one cannot choose $a$ to be a non-square.) | |
| Nov 21, 2014 at 23:58 | review | First posts | |||
| Nov 22, 2014 at 1:17 | |||||
| Nov 21, 2014 at 23:47 | history | edited | Tomo | CC BY-SA 3.0 |
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| Nov 21, 2014 at 23:40 | comment | added | Tomo | Thanks for comments. The trivial subgroup is not what I want. I am looking for positive dimensional subgroups. Also, I would like to know what happens if k is algebraically closed field of characteristic 2? Are there any over/sub groups of $H$ in $PGL_4(\bar k)$? | |
| Nov 21, 2014 at 23:39 | history | edited | Tomo | CC BY-SA 3.0 |
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| Nov 21, 2014 at 17:26 | comment | added | Nick Gill | Being pedantic: I guess the trivial subgroup answers the second part of your question. | |
| S Nov 21, 2014 at 17:09 | history | suggested | Seirios | CC BY-SA 3.0 |
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| Nov 21, 2014 at 17:02 | review | Suggested edits | |||
| S Nov 21, 2014 at 17:09 | |||||
| Nov 21, 2014 at 14:39 | comment | added | Nick Gill | Do $w$ and $z$ vary across $k$ too? | |
| Nov 21, 2014 at 8:12 | history | edited | Tomo | CC BY-SA 3.0 |
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| Nov 21, 2014 at 7:35 | comment | added | Tomo | Note that $H$ is a group. | |
| Nov 21, 2014 at 7:34 | history | asked | Tomo | CC BY-SA 3.0 |