Exercise 13.1.4

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\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}}

Use Example 7.3.4 from Chapter 7 to show that (13.8) gives all subgroups of $\langle (1\,3\,2\,4), (1\,2)angle$ of order 4 or 8.

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} We obtain all subgroups of $D_8 \simeq \langle \sigma, au angle$, where $\sigma = (1\,3\,2\,4), au = (1\,2)$, in Exercise 7.3.3 \begin{center} \begin{tikzpicture} ode (n0) at (7,6) {$\{e\}$}; ode (n1) at (1,4) {$\langle au angle$}; ode (n2) at (4,4) {$\langle \sigma^2 au angle$}; ode (n3) at (7,4) {$\langle\sigma^2 angle$}; ode (n4) at (10,4) {$\langle \sigma au angle$}; ode (n5) at (13,4) {$\langle \sigma^3 au angle$}; ode (n6) at (4,2) {$\langle \sigma^2, au angle$}; ode (n7) at (7,2) {$\langle \sigma angle$}; ode (n8) at (10,2) {$\langle \sigma^2, \sigma au angle$}; ode (n9) at (7,0) {$G = \langle \sigma, au angle$}; \draw[->] (n0) edge (n3) edge (n2) edge (n1) edge (n4) edge (n5); \draw[<-] (n6) edge (n1) edge (n2) edge (n3); \draw[<-] (n8) edge (n3) edge (n4) edge (n5); \draw[->] (n3) edge (n7); \draw[<-] (n9) edge (n6) edge (n7) edge (n8); \end{tikzpicture} \end{center} If $G$ is a subgroup of order 4 or 8, then $G$ is one of the four groups $$\langle \sigma^2, au angle, \quad \langle \sigma angle,\quad \langle \sigma^2, \sigma au angle, \quad \langle \sigma, au angle,$$ Moreover $\sigma^2 = (1\,2)(3\,4)$ and $\sigma au = (1\,4)(2\,3)$, so $$\langle \sigma^2, au angle = \langle (1\,2)(3\,4), (1\,2) angle = \langle (3\,4), (1\,2) angle,$$ and $$\langle \sigma^2, \sigma au angle = \langle (1\,2)(3\,4), (1\,4)(2\,3) angle = \langle (1\,2)(3\,4), (1\,3)(2\,4) angle$$ is the group of double transpositions $\{(), (1\,2)(3\,4), (1\,4)(2\,3), (1\,3)(2\,4)\}$. Therefore $G$ is one of the four groups given in the text $$\langle (1\,2),(3\,4) angle,\qquad \langle (1\,2)(3\,4), (1\,3)(2\,4) angle,\qquad \langle (1\,3\,2\,4) angle, \qquad \langle (1\,3\,2\,4), (1\,2) angle.$$ \end{proof}

Exercise 13.1.4

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Exercise 13.1.4

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