Exercise 7.4.5

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}}

Label the elements of $S_3$ as $g_1 = e,g_2 = (1\,2\,3),g_3 = (1\,3\,2),g_4 = (1\,2),g_5 = (1\,3)$, and $g_6 = (2\,3)$. Write down the six permutations $\sigma_i\in S_6$ defined by the rows of the Cayley table (7.18).

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 numbering of $S_3$ is given by $$g_1=e,g_2 = (1 2 3), g_3 = (1 3 2), g_4 = (1 2), g_5 = (1 3), g_6 = (2 3).$$
Write $\sigma_i$ the permutation defined by $g_i g_j = g_{\sigma_i(j)}, \ 1 \leq i,j \leq n$.
The Cayley table of the group gives
$$
\left[
\begin{array}{cccccc}
g_1 & g_2 &  g_3 & g_4 & g_5  & g_6\
g_2 & g_3 &  g_1 & g_5 & g_6  & g_4\
g_3 & g_1 &  g_2 & g_6 & g_4  & g_5\
g_4 & g_6 &  g_5 & g_1 & g_3  & g_2\
g_5 & g_4 &  g_6 & g_2 & g_1  & g_3\
g_6 & g_5 &  g_4 & g_3 & g_2  & g_1\
\end{array}
ight]
$$
where the element of the $i$th row, $j$th column is $g_i \circ g_j = g_i g_j =  g_{\sigma_i(j)}$.

Thus
\begin{align*} 
\sigma_1 &= (),\
\sigma_2 &= (1 2 3)(4 5 6),\
\sigma_3 &=(1 3 2)(4 6 5) = \sigma_2^2,\
\sigma_4 &= (1 4)(2 6)(3 5),\
\sigma_5 &= (1 5)(2 4)(3 6),\
\sigma_6 &= (1 6)(2 5)(3 4).
\end{align*}
\end{proof}

Exercise 7.4.5

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Exercise 7.4.5

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