Timeline for Why inherit the Tits systems structure by a $B$-adapted homomorphism?
Current License: CC BY-SA 4.0
9 events
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| Jan 12, 2022 at 14:18 | comment | added | LSpice |
TeX note: please use, e.g., $\operatorname{Ker} \phi$ \operatorname{Ker} \phi instead of $\rm Ker \phi$ \rm Ker \phi; particularly note the spacing.
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| Jan 12, 2022 at 14:16 | history | edited | LSpice | CC BY-SA 4.0 |
Proofreading
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| Jan 12, 2022 at 11:55 | comment | added | M masa | Now I proved a part of 2. The intersection $\phi(N)\cap \hat{B}$ is normal in $\phi(N)$. For any $x,y\in N$ such that $\phi(x)\in \hat{B}$, we have $\phi(xBx^{-1})=\phi(B)$ from the definition ${\rm Stab} B$. So we have $\phi(xb_{1}x^{-1}b_{2})=0$ for any $b_{1}\in B$ and some $b_{2}\in B$. Since ${\rm Ker}\phi\subset B$, $xb_{1}x^{-1}b_{2}\in B$. As $N_{G}(B)=B$, this means that $x\in B\cap N (\triangleleft N)$. Then we have $\phi(yxy^{-1})\in \hat{B}\cap \phi(N)$. Thus $\hat{B}\cap \phi(N)\triangleleft \phi(N)$. | |
| Jan 12, 2022 at 10:48 | history | edited | M masa | CC BY-SA 4.0 |
added 44 characters in body
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| Jan 12, 2022 at 10:00 | comment | added | M masa | I can prove 1. For any $g\in \hat{G}$, there exists $h\in G$ such that $\phi(h^{-1})g\in {\rm Stab} B$. Since $\phi(G)\subset \hat{G}_{0}$, we have $\hat{G}=\hat{G}_{0}\cdot {\rm Stab} B$. Thus the restriction of $\xi$ to ${\rm Stab} B$ is surjective. | |
| Jan 12, 2022 at 9:12 | comment | added | M masa | Thank you. It's a careless mistake. | |
| Jan 12, 2022 at 9:01 | history | edited | M masa | CC BY-SA 4.0 |
added 14 characters in body
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| Jan 12, 2022 at 8:14 | comment | added | HJRW | Should the equation concerning $\phi(C(w))$ mention $\xi$ somewhere? | |
| Jan 11, 2022 at 12:32 | history | asked | M masa | CC BY-SA 4.0 |