Timeline for About the conjugation of semi-simple subgroups
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 22, 2016 at 8:51 | comment | added | YCor | @MikhailBorovoi yes; for better bounds, see mathoverflow.net/questions/168292 (it's $n!2^n$ for large $n$). | |
| May 22, 2016 at 4:29 | comment | added | Mikhail Borovoi | @YCor: Thank you! Indeed, the congruence subgroup modulo 3 has no elements of finite order, hence any finite subgroup of $GL_n(\mathbf{Z})$ embeds into $GL_n(\mathbf{F}_3)$ and therefore, it is of order $\le 3^{n^2}$. | |
| May 21, 2016 at 21:39 | comment | added | YCor | @MikhailBorovoi: see mathoverflow.net/questions/71969, namely Robinson's answer and its comments: some congruence subgroup is torsion-free. So the index is an upper bound on the order of a finite subgroup. | |
| May 21, 2016 at 16:00 | comment | added | Mikhail Borovoi | @YCor: You write that $GL_n(\mathbf{Z})$ has finite subgroups of bounded order. Could you please give a reference? | |
| May 18, 2016 at 12:16 | vote | accept | Golden Wave | ||
| May 17, 2016 at 19:48 | answer | added | Mikhail Borovoi | timeline score: 9 | |
| May 16, 2016 at 23:52 | comment | added | YCor | If it helps, $N$ is reductive with a bounded number of connected components, and can be supposed to be split. | |
| May 16, 2016 at 20:06 | comment | added | Mikhail Borovoi | The answer seems to be YES. We may pass to a finite extension $k$ of bounded degree. By the comment of @YCor we may assume that $H_1$ and $H_2$ are $k$-isomorphic. After we choose a conjugating element $g\in G( \bar{k})$, we obtain a cohomology class in $H^1(k,N)$, where $N=N(H_1)$ is the normalizer of $H_1$ in $G$. It remains to show that a cohomology class in $H^1(k,N)$ can be killed by a field extension of bounded degree. | |
| May 16, 2016 at 19:13 | comment | added | YCor | Since $GL_n(\mathbf{Z})$ has finite subgroups of bounded order, you already know that there exists $d(k)$ such that every semisimple group of dimension $\le k$ splits over an extension of degree $\le d(k)$. Of course this is weaker than what you ask. | |
| May 16, 2016 at 19:02 | history | edited | Golden Wave | CC BY-SA 3.0 |
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| May 16, 2016 at 18:39 | history | asked | Golden Wave | CC BY-SA 3.0 |