Timeline for The group of automorphisms a pair $(G,X)$ where $X$ is a spherical homogeneous space of $G$
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Apr 15, 2017 at 20:39 | comment | added | Mikhail Borovoi | @YCor: I think that $Z(G)$ maps into ${\rm Aut}_G(X)$ and that the kernel of $\bar\pi$ (with the notation of my edit) is the quotient of ${\rm Aut}_G(X)=N(H)/H$ by the image of $Z(G)$, that is, $N(H)/(Z(G)H)$. | |
| Apr 15, 2017 at 19:08 | comment | added | YCor | You're right, I wasn't careful. | |
| Apr 15, 2017 at 17:07 | comment | added | Mikhail Borovoi | @YCor: Concerning your second comment, I do not agree with the assertion that $Z_X(G)\subset {\rm Aut}_G(X)$ (this assertion is implicit in the formula ${\rm Aut}_G(X)/Z_X(G)\ $). See my edit. | |
| Apr 15, 2017 at 15:54 | history | edited | Mikhail Borovoi | CC BY-SA 3.0 |
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| Apr 15, 2017 at 15:09 | history | edited | Mikhail Borovoi | CC BY-SA 3.0 |
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| Apr 15, 2017 at 14:38 | comment | added | Mikhail Borovoi | @YCor: I agree with the first comment; I have edited my post. | |
| Apr 15, 2017 at 14:32 | history | edited | Mikhail Borovoi | CC BY-SA 3.0 |
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| Apr 15, 2017 at 11:42 | comment | added | YCor | You have defined a subgroup $\mathrm{Inn}(G,X)$ (the set of all $i_a$ when $a$ ranges over $G$); is by definition $\mathfrak{S}$ the quotient of $\mathrm{Out}(G,X)=\mathrm{Aut}(G,X)/\mathrm{Inn}(G,X)$? If so I don't see why you need any splitting to define the extension. Also ther kernel of $G\to\mathrm{Inn}(G,X)$ consists in the intersection $Z_X(G)$ of the kernel of the $G$-action on $X$ with the center of $G$. This yields an extension $1\to\mathrm{Aut}_G(X)/Z_X(G)\to\mathrm{Out}(G,X)\to\mathrm{Out}_X(G)\to 1$. | |
| Apr 15, 2017 at 11:15 | comment | added | YCor | The kernel of $\pi$ is rather those $\mu$ such that $\mu(g.x)=g\cdot\mu(x)$ for all $g,x$, isn't it? (The unit element of $Aut(G)$ is not the constant 1, it is the identity map.) It seems to match the sequel of your post. | |
| Apr 15, 2017 at 10:45 | history | edited | Mikhail Borovoi | CC BY-SA 3.0 |
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| Apr 15, 2017 at 10:39 | history | asked | Mikhail Borovoi | CC BY-SA 3.0 |