Exercise 13.2.4

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 (13.19) gives coset representatives of $\mathrm{AGL}(1,\F_5)$ in $S_5$.

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}
As the index $(S_5:\mathrm{AGL}(1,\F_5)) =120/20 = 6$, it is sufficient to verify that the 6 permutations 
$$S = \{e, (1\,2\,3), (2\,3\,4), (3\,4\,5), (1\,4\,5),(1\,2\,5)\}$$
are in distinct coset, by verifying that the 15 permutations $u v^{-1} 
ot \in \mathrm{AGL}(1,\F_5)$, where $$u,v \in S,\qquad u 
e v.$$

\begin{align*}
\mathrm{AGL}(1,\F_5) &= \{e, (1,2,4,3), (1,2,3,4,5), (1,3,5,2,4), (1,4,5,2), (1,3,2,5),\
&(1,4)(2,3), (1,4,2,5,3), (2,5)(3,4), (1,5,3,4), (2,3,5,4), (1,3)(4,5),\
&(1,3,4,2), (1,5)(2,4), (1,2)(3,5), (1,5,4,3,2), (2,4,5,3), (1,4,3,5),\
&(1,2,5,4), (1,5,2,3) \}
\end{align*}

\begin{align*}
&\{u v^{-1} \ | \ u \in S, v \in S, u
e v\} =\
&\{ (1,3,2), (2,4,3), (3,5,4), (1,5,4), (1,5,2), (1,2,3,5,4), (2,3,5),\
&(1,3,4), (1,5,4,2,3), (1,5,2,3,4), (1,3,4,5,2), (2,4,5), (1,5,3),\
&(1,5,3,4,2), (1,2,4)\}\
\end{align*}

Sage instructions
\begin{verbatim}
S5 = SymmetricGroup(5)
a = S5([(1,2,3,4,5)])
b = S5([(1,2,4,3)])
G = PermutationGroup([a,b])
l =  [S5([]),S5([(1,2,3)]),S5([(2,3,4)]),S5([(3,4,5)]),S5([(1,4,5)]),S5([(1,2,5)])]
[u*v^(-1) in G for u in l for v in l if u< v]
\end{verbatim}
\begin{center}
[False, False, \ldots , False]
\end{center}

So $S$ is a set of coset representatives of $\mathrm{AGL}(1,\F_5)$ in $S_5$.
\end{proof}

Exercise 13.2.4

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Exercise 13.2.4

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