Exercise 1.7.11 (Action of $D_8$ on the set of vertices of a square)

Question

Write out the cycle decomposition of the eight permutations in $S_4$ corresponding to the elements of $D_8$ given by the action of $D_8$ on the vertices of a square (where the vertices of the square are labelled as in Section 2).

Richard Ganaye · Accepted Answer

\begin{proof} 
 The vertices of the square are labelled as in Section 2 (p. 24).

We know that 
$$D_8 = \{e, r, r^2, r^3, s, rs, r^2s, r^3s\},$$
where $r$ is the rotation about the center through $\pi/2$ radian, and $s$ the reflection through the $x$-axis (for instance).

Then the action of $D_8$ on the vertices $1,2,3,4$ are given in the following array ( using $r(1) = 4,\, r(2) = 1,\, r(3) = 2,\, r(4) = 3,  $ and $s(1) = 2,\, s(2) = 1,\, s(3) = 4,\, s(4) = 3$)

\begin{center}
\begin{tabular}{|l|cccc|c| }
\hline
 &1& 2  & 3 & 4 & dec.\
\hline
 $e$ & 1& 2  & 3  & 4&()\
 $r$ & 4 & 1 & 2 & 1 &(1\ 4 \ 3\ 2)\
 $r^2$ & 3& 4  & 1 & 2 &(1\ 3)(2\ 4)\
 $r^3$ & 2& 3 & 4 & 1&(1\ 2 \ 3 \ 4)\
 $s$ & 2& 1 &4  &3& (1\ 2)(3\ 4)\
 $r s$ &1& 4 & 3 &2&(2\ 4) \
 $r^2 s$ &4& 3 & 2 & 1& (1\ 4)(2\ 3)\
 $r^3s$ & 3& 2 &  1& 4&(1\ 3)\
\hline
\end{tabular}
\end{center}
\end{proof}
Note: Since the action of $D_8 = \langle r^3, sangle$ on the set of vertices of the square is faithful (see Example 4), this shows that
$$D_8 \simeq \langle (1\ 2 \ 3 \ 4), (1\ 2)(3\ 4) angle \subset S_4.$$
(Alternatively, $D_8 \simeq  \langle (1\ 2 \ 3 \ 4), (1\ 3) angle$)

\bigskip
Verification with sagemath:
\begin{verbatim}
sage: G = PermutationGroup([[(1,2,3,4)], [(1,2),(3,4)]])
sage: G
Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]
sage: G.cardinality()
8
sage: list(G)
[(), (1,2)(3,4), (1,2,3,4), (1,3), (1,3)(2,4), (2,4), (1,4,3,2), (1,4)(2,3)]
\end{verbatim}


	1	2	3	4	dec.

$e$	1	2	3	4	()
$r$	4	1	2	1	(1 4 3 2)
$r^{2}$	3	4	1	2	(1 3)(2 4)
$r^{3}$	2	3	4	1	(1 2 3 4)
$s$	2	1	4	3	(1 2)(3 4)
$rs$	1	4	3	2	(2 4)
$r^{2} s$	4	3	2	1	(1 4)(2 3)
$r^{3} s$	3	2	1	4	(1 3)

Exercise 1.7.11 (Action of $D_8$ on the set of vertices of a square)

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Exercise 1.7.11 (Action of $D_8$ on the set of vertices of a square)

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