Timeline for To what extent does a torsor determine a group
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 8, 2018 at 17:04 | comment | added | მამუკა ჯიბლაძე | @JulianRosen I've now asked a question about this, and a comment by Laurent Moret-Bailly there provides useful information about it: if $G$ and $G'$ are locally isomorphic, then the object $\operatorname{\underline{Iso}}(G',G)$ of isomorphisms between them is an $\operatorname{\underline{Aut}}(G)$-torsor; and if there is a $G$-$G'$-bitorsor, then $\operatorname{\underline{Iso}}(G',G)$ must be extended from a $G$-torsor along the canonical homomorphism $G\to\operatorname{\underline{Aut}}(G)$. | |
| Jan 8, 2018 at 3:43 | comment | added | მამუკა ჯიბლაძე | Btw this implies that $G$ and $G'$ become isomorphic over $T$, since for any given $e\in T$ there is an isomorphism between $m$ and $m'$ sending $x$ to $m(e,x,e)$; however over $T$ in fact $T$ trivializes both over $G$ and $G'$, so this is trivial anyway. I believe there must be examples when $G$ and $G'$ are not isomorphic. | |
| Jan 7, 2018 at 6:24 | comment | added | მამუკა ჯიბლაძე | @JulianRosen Correct. That one is $m'(x,y,z)=m(z,y,x)$ | |
| Jan 7, 2018 at 4:27 | comment | added | Julian Rosen | I think I see. The right action of $G'$ can be turned into a left action by composing with inversion, and my confusion came from the fact that this makes $T$ into both a (left) $G$-torsor and a (left) $G'$-torsor. If I understand correctly, this is not a contradiction because the heap structure coming from the left $G'$-action is not the heap structure $T$ starts with. | |
| Jan 7, 2018 at 2:00 | comment | added | David Roberts♦ | @Julian a left G torsor and right G' torsor, in a compatible way. | |
| Jan 7, 2018 at 1:41 | comment | added | Julian Rosen | I am confused about one point. As you say, $T$ is a $G-G'$-bitorsor. Doesn't this mean $T$ is both a torsor for $G$ and a torsor for $G'$? | |
| Jan 6, 2018 at 22:04 | history | answered | მამუკა ჯიბლაძე | CC BY-SA 3.0 |