Exercise 13.3.11

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}}

Prove that  $\mathrm{GL}(3,\mathbb{F}_2)=\mathrm{SL}(3,\mathbb{F}_2) \simeq \mathrm{PGL}(3,\mathbb{F}_2)=\mathrm{PSL}(3,\mathbb{F}_2)$.

\begin{proof}
A matrix $M \in \mathrm{GL}(3,\mathbb{F}_2)$ is such that $\det(M)$ is invertible in $\F_2$, but only 1 is invertible in $\F_2$, so that $\det(M) = 1$ and $M \in \mathrm{SL}(3,\mathbb{F}_2)$. Therefore $\mathrm{GL}(3,\mathbb{F}_2) \subset \mathrm{SL}(3,\mathbb{F}_2)$, and $ \mathrm{SL}(3,F) \subset \mathrm{GL}(3,F)$ is true in every field $F$, thus
$$\mathrm{GL}(3,\mathbb{F}_2)=\mathrm{SL}(3,\mathbb{F}_2).$$

By definition, in any field $F$, $\mathrm{PGL}(n,F)=\mathrm{GL}(n,F)/F^*I$ and $\mathrm{PSL}(n,F)=\mathrm{SL}(n,F)/(F^*I\cap \mathrm{SL}(n,F))$. When $F=\mathbb{F}_2$, then $\mathbb{F}_2^*=\{1\}$, i.e., $\mathbb{F}_2^*I=(\mathbb{F}_2^*I\cap \mathrm{SL}(n,\mathbb{F}_2))=\{I\}$. Therefore, for n=3,
$$\mathrm{GL}(3,\mathbb{F}_2)=\mathrm{SL}(3,\mathbb{F}_2) \simeq \mathrm{PGL}(3,\mathbb{F}_2)=\mathrm{PSL}(3,\mathbb{F}_2).$$
\end{proof}

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}}

Exercise 13.3.11

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Exercise 13.3.11

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