Exercise 4.2.6 (A not faithful action of $D_8$ on $S_4$)

Question

Let $r$ and $s$ be the usual generators for the dihedral group of order $8$ and let $N = \langle r^2\rangle$. List the left cosets of $N$ in $D_8$ as $1N, rN, s N$ ans $sr N$. Label these cosets with the integers $1,2,3,4$ respectively. Exhibit the image of each element of $D_8$ under the representation $\pi_N$ of $D_8$ into $S_4$ obtained from the action of $D_8$ by left multiplication on the set of $4$ left cosets of $N$ in $D_8$. Deduce that this representation is not faithful and prove that $\pi_N(D_8)$ is isomorphic to the Klein $4$-group.

Richard Ganaye · Accepted Answer

\begin{proof} The left cosets of $N = \langle r^2angle = \{1,r^2\}$ are
$$1N = \{1,r^2\}, \qquad rN = \{r,r^3\},\qquad sN = \{s,sr^2\},\qquad sr N = \{sr, sr^3\}.$$
We label these cosets with the bijection $f$:
\begin{center}
\begin{tabular}{c|cccc}
$k$ & $1$ & $2$ & $3$ & $4$ \
\hline
$f(k)$ & $1N$ &$rN$& $sN$ &$ srN$
\end{tabular}
\end{center}
We write $\sigma = \pi_N$ the associate representation. Then
\begin{align*}
s\cdot 1N = sN & \Rightarrow \sigma_s(1) = 3,\
s\cdot rN =  sr N & \Rightarrow \sigma_s(2) = 4,\
s\cdot sN = 1N & \Rightarrow \sigma_s(3) = 1,\
s \cdot srN = rN & \Rightarrow \sigma_s(4) = 2,
\end{align*}
so $\sigma_s = (1\ 3)(2\ 4)$.

Moreover
\begin{align*}
r\cdot 1N =rN & \Rightarrow \sigma_r(1) = 2,\
r\cdot rN =  \{r^2,1\} = 1N& \Rightarrow \sigma_r(2) = 1,\
r\cdot sN = \{rs,rsr^2\} = \{sr^3, sr\} = sr N& \Rightarrow \sigma_r(3) = 4,\
r \cdot srN = sN & \Rightarrow \sigma_r(4) = 3,
\end{align*}
so $\sigma_r = (1\ 2)(3\ 4)$.

Since the representation $\pi_N = \sigma : g \mapsto \sigma_g$ is an homomorphism, we obtain
\begin{align*}
\sigma_e &= (),\
\sigma_r &=  (1\ 2)(3\ 4),\
\sigma_{r^2} &= (),\
\sigma_{r^3} &= (1\ 2)(3\ 4),\
\sigma_s &= (1\ 3)(2\ 4),\
\sigma_{sr} &= (1\ 4)(2\ 3),\
\sigma_{sr^2} &= (1\ 3)(2\ 4),\
\sigma_{sr^3} &= (1\ 4)(2\ 3).
\end{align*}
Therefore $ \ker(\pi_N) = \{e,r^2\}$, so this representation is not faithful. The image $\pi_N(D_8)$ of the representation $\sigma = \pi_N$ is
\begin{align*}
\mathrm{im}(\pi_N) &= \{(),(1\ 2)(3\ 4), (1\ 3)(2\ 4), (1\ 4)(2\ 3)\}\
&\simeq V_4 = Z_2	imes Z_2,
\end{align*}
since every element has order $1$ or $2$.

This representation affords the homomorphism 
$$D_8/\langle r^2 angle \simeq Z_2 	imes Z_2.$$
\end{proof}

Exercise 4.2.6 (A not faithful action of $D_8$ on $S_4$)

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$k$	$1$	$2$	$3$	$4$

$f (k)$	$1 N$	$𝑟𝑁$	$𝑠𝑁$	$𝑠𝑟𝑁$

Exercise 4.2.6 (A not faithful action of $D_8$ on $S_4$)

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