Exercise 13.1.16

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}}

Consider the subgroups $\langle (1\,2),(3\,4) angle$ and $\langle (1\,2)(3\,4),(1\,3)(2\,4) angle$ of $S_4$.
\be
\item[(a)] Prove that these subgroups are isomorphic but not conjugate. This shows that when classifying subgroups of a given group, it can happen that nonconjugate subgroups can be isomorphic as abstract groups.
\item[(b)] Explain why the subgroup $\langle (1\,2),(3\,4) angle$ isn't mentioned in Theorems 13.1.1 and 13.1.6.
\ee

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}
\be
\item[(a)]
\begin{align*}
H_1 &= \langle (1\,2),(3\,4) angle = \{(), (1\,2),(3\,4), (1\,2) (3\,4)\}, \
 H_2 &= \langle (1\,2)(3\,4),(1\,3)(2\,4) angle = \{ (), (1\,2)(3\,4),(1\,3)(2\,4), (1\,4)(2\,3)\}
 \end{align*}
  are both isomorphic to the Klein's group $\Z/2\Z 	imes \Z/2\Z$.
  
Every conjugate of $(1\,2) \in H_1$ by $\sigma \in S_4$ is $(\sigma(1)\, \sigma(2)) $, which is not in $H_2$. The subgroups $H_1,H_2$ are not conjugate. 
\item[(b)] $H_1 = \langle (1\,2),(3\,4) angle$ is not a transitive subgroup of $S_4$ (the orbit of 1 is $\{1,2\}$), so isn't mentioned in Theorems 13.1.1 and 13.1.6.
\ee
\end{proof}

\subsection{QUINTIC POLYNOMIALS}

Exercise 13.1.16

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0.1 QUINTIC POLYNOMIALS

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Exercise 13.1.16

Answers

0.1 QUINTIC POLYNOMIALS

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