Exercise 6.4.9

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}}

Explain how (6.6) and (6.10) relate to the last paragraph of the discussion of semidirect products in the Mathematical Notes.

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}
The group homomorphism $\psi$ of Exercise 8 shows that $(Htimes G) / (H 	imes\{e\}) \simeq G$.

Moreover the isomorphism (6.10)
$
\phi : 
\left\{
\begin{array}{ccc}
  \mathrm{AGL}(1,\F_p)  &   	o & \F_p times \F_p^*  \
 \gamma_{a,b}& \mapsto   &  (b,a) 
\end{array}
ight.
$

maps $T = \{\gamma_{1,b} \ \vert \ b \in \F_p\}$ on $\F_p 	imes \{1\}$.
 
Therefore $\mathrm{AGL}(1,\F_p)/T \simeq (\F_p times \F_p^*)/(\F_p 	imes \{1\}) \simeq \F_p^*$ : we obtain so (6.6) :
 \begin{center}
  $\mathrm{AGL}(1,\F_p)/T \simeq \F_p^*$.
\end{center}
\end{proof}

Exercise 6.4.9

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Exercise 6.4.9

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