Exercise 14.3.24

Question

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}% intervalles d'entiers 
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\be} {\begin{enumerate}}

ewcommand{\ee} {\end{enumerate}}

ewcommand{\deb}{\begin{eqnarray*}}

ewcommand{\fin}{\end{eqnarray*}}

ewcommand{\ssi} {si et seulement si }

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{\U}{\mathbb{U}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\im}{\,\mathrm{Im}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{\Gal}{\mathrm{Gal}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

The action of $\mathrm{PGL}(2,F)$ on $\hat F = F \cup \{\infty\}$ was introduced in Section 7.5 In particular, Exercise 11 of that section implies that the isotropy subgroup of $\mathrm{PGL}(2,F)$ at the point $\infty$ can be identified with $\mathrm{AGL}(1,F)$. Use part (c) of Exercise 4 and Exercise 19 to prove that the action of $\mathrm{PGL}(2,F)$ on $\hat F$ is 3-transitive (also called triply transitive).

Richard Ganaye · Accepted Answer

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}% intervalles d'entiers 
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\be} {\begin{enumerate}}

ewcommand{\ee} {\end{enumerate}}

ewcommand{\deb}{\begin{eqnarray*}}

ewcommand{\fin}{\end{eqnarray*}}

ewcommand{\ssi} {si et seulement si }

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{\U}{\mathbb{U}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\im}{\,\mathrm{Im}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{\Gal}{\mathrm{Gal}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

\begin{proof}
By Exercise 4 (c), $\mathrm{GL}(2,F)$ acts transitively on $F^2 \setminus \{0\}$. This proves that $\mathrm{PGL}(2,F)$ acts transitively on the projective line $\mathbb{P}_1(F)$, so that the action of $\mathrm{PGL}(2,F)$ on $\hat F$ is transitive.

This allows us to apply Exercise 19 (b). To prove that the action of $\mathrm{PGL}(2,F)$ on $\hat F$ is triply transitive, it is sufficient to prove that the isotropy group $G_{\infty} = \mathrm{AGL}(1,F)$ acts 2-transitively on $ \hat F \setminus \{\infty\} = F$. Example 14.3.2 shows that this is true. Therefore the action of $\mathrm{PGL}(2,F)$ on $\hat F$ is 3-transitive.
\end{proof}

Exercise 14.3.24

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Exercise 14.3.24

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