Timeline for Finite subgroups of $\mathrm{O}_n(\mathbb R)$ from finite subgroups of $\mathrm{SO}_n(\mathbb R)$
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Oct 15, 2023 at 9:09 | history | edited | Dave Benson | CC BY-SA 4.0 |
Expanded the argument a bit.
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| Oct 13, 2023 at 23:14 | comment | added | Andrea Aveni | I see, thank you! | |
| Oct 13, 2023 at 23:13 | vote | accept | Andrea Aveni | ||
| Oct 13, 2023 at 23:03 | comment | added | YCor | The normalizer $N$ of $G$ in $\mathrm{SO}_n$ is reduced to $G$. The reason is that $N/G$ being a compact Lie group, if it is not trivial, then it has a nontrivial finite subgroup, and hence pulling back contradicts the maximality of $G$ among finite subgroups of $\mathrm{SO}_n$. Hence either $H$ is empty and $G$ equals its normalizer in $\mathrm{O}_n$, or $H$ is nonempty and $G$ has index two in its normalizer, which is then $G\sqcup H$. | |
| Oct 13, 2023 at 20:47 | comment | added | Andrea Aveni | Sorry, but why is that $\ker \det|_{N_{\mathrm{O}_n(\mathbb R)}(G)}=G$ and not some dense subgroup of $\mathrm{SO}_n(\mathbb R)$? I guess this equality holds iff $N_{\mathrm{SO}_n(\mathbb R)}(G)$ is finite, but I don't see why this must be the case. | |
| Oct 13, 2023 at 19:17 | comment | added | Max Horn | For index reasons: restrict the determinant to the normalizer $N$; then its kernel must be $G$, and by the homomorphism theorem the kernel has index 2. | |
| Oct 13, 2023 at 19:04 | comment | added | Andrea Aveni | I agree that $G^h=G$ and thus $\langle G,h\rangle$ is a subgroup of the normalizer of $G$ in $\mathrm O_n(\mathbb R)$. I also agree that if we assume this normalizer to be finite, then $\langle G,h\rangle=\langle G, h'\rangle$ (because otherwise $hh'$ would be in $\mathrm{SO}_n(\mathbb R)\setminus G$ and, with $G$ it would generate an infinite group). But why should this normalizer be finite? Thank you | |
| Oct 13, 2023 at 16:47 | history | edited | Dave Benson | CC BY-SA 4.0 |
added 233 characters in body
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| Oct 13, 2023 at 16:36 | history | answered | Dave Benson | CC BY-SA 4.0 |