Timeline for Which reflection groups can be enlarged?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 12, 2021 at 2:15 | comment | added | Richard Lyons | $B_3=Z\times S$ where $|Z|=2$ (generated by the -1 mapping) and $S=D_3\cong S_4$, the symmetric group. There is a unique surjective homomorphism $\sigma:S\to Z$, the sign homomorphism. Then the mapping $\alpha:(z,s)\mapsto (z\sigma(s),s)$ is an automorphism of $Z\times S\cong B_3$. There are reflections in $S$, and they are carried by $\alpha$ to their negatives, so $\alpha$ is not inner -- it's not even conjugation by any element of $GL_3(R)$. | |
| Feb 12, 2021 at 1:06 | comment | added | M. Winter | @RichardLyons Can you say more about this? | |
| Feb 12, 2021 at 0:50 | comment | added | Will Sawin | @RichardLyons Good point, thanks. | |
| Feb 12, 2021 at 0:50 | history | edited | Will Sawin | CC BY-SA 4.0 |
added 15 characters in body
|
| Feb 12, 2021 at 0:41 | comment | added | Richard Lyons | I believe B_3 does have an outer automorphism, but not in $O(R^3)$. | |
| Feb 11, 2021 at 16:17 | history | answered | Will Sawin | CC BY-SA 4.0 |