Exercise 2.5.6 (Centralizers of every element in $D_8$, $Q_8$, $S_3$, $D_{16}$)

Question

Use the given lattices to help to find the centralizers of every element in the following groups:
$$	ext{\bf(a)}\  D_8 \qquad 	ext{\bf(b)}\  Q_8 \qquad	ext{\bf(c)}\  S_3 \qquad	ext{\bf(d)}\  D_{16}. $$

Richard Ganaye · Accepted Answer

\begin{proof} Centralizers of every element in $D_8,\, Q_8,\, S_3,\, D_{16}$.
\begin{enumerate}
\item[{\bf(a)}] $G = D_8$.

For instance, The centralizer $C_{D_8}(s)$ contains $s$ and $r^2$, so contains $\langle s, r^2 angle$ Since $r 
ot \in C_{D_8}(s)$, we obtain
$$\langle s, r^2 angle \leq C_{D_8}(s) < D_8.$$
Then the diagram (4) shows that 
$$ C_{D_8}(s) = \langle s, r^2 angle.$$
More generally, we obtain the following array
\begin{center}
\begin{tabular}{|c|cccccccc|}
\hline
 $x$  & $1$ & $r$ & $r^2$  & $r^3$ & $s$ & $rs$ & $r^2 s$ & $r^3 s$ \
\hline
 $C_G(x)$&$D_8$  &$\langle r angle$  & $D_8$  & $\langle r angle$ & $\langle s, r^2 angle$ & $\langle rs, r^2 angle$ & $\langle s, r^2 angle$ & $\langle rs, r^2 angle$ \
\hline
\end{tabular}
\end{center}

\item[{\bf(b)}] $G = Q_8.$

Similarly

\begin{center}
\begin{tabular}{|c|cccccccc|}
\hline
 $x$  & $1$ & $i$ & $j$  & $k$ & $-1$ & $-i$ & $-j$ & $-k$ \
\hline
 $C_G(x)$&$Q_8$  &$\langle  i angle$  & $\langle  jangle$  & $\langle kangle$ & $Q_8$ &$\langle  i angle$  & $\langle  jangle$  & $\langle kangle$ \
\hline
\end{tabular}
\end{center}

\item[{\bf(c)}] $G = S_3$.
\begin{center}
\begin{tabular}{|c|cccccc|}
\hline
 $x$  & $()$ & $(1\ 2)$ & $(1 \ 3)$  & $(2\ 3)$ & $(1\ 2 \ 3)$ & $(1\ 3\ 2)$  \
\hline
 $C_G(x)$&$S_3$  &$\langle (1\ 2) angle$  & $\langle  (1 \ 3)angle$  & $\langle (2\ 3) angle$ & $\langle (1\ 2\ 3) angle$ &$\langle (1\ 2\ 3) angle$  \
\hline
\end{tabular}
\end{center}

\item[{\bf(d)}] $G = D_{16}.$

\begin{center}
\begin{tabular}{|c|cccccccc|}
\hline
 $x$  & $1$ & $r$ & $r^2$  & $r^3$ & $r^4$ & $r^5$ & $r^6$ & $r^7$ \
\hline
 $C_G(x)$&$D_{16}$  &$\langle  r angle$  & $\langle  rangle$  & $\langle rangle$ & $D_{16}$ &$\langle  r angle$  & $\langle  r angle$  & $\langle rangle$ \
\hline
\end{tabular}
\end{center}

\begin{center}
\begin{tabular}{|c|cccccccc|}
\hline
 $x$  & $s$ & $sr$ & $sr^2$  & $sr^3$ & $sr^4$ & $sr^5$ & $sr^6$ & $sr^7$ \
\hline
 $C_G(x)$&$\langle  s, r^4angle$  &$\langle  sr, r^4 angle$  & $\langle  sr^2, r^4 angle$  & $\langle  sr^3, r^4angle$ & $\langle  s, r^4 angle$ &$\langle sr, r^4 angle$  & $\langle  sr^2, r^4angle$  & $\langle sr^3, r^4angle$ \
\hline
\end{tabular}
\end{center}

\end{enumerate}

\end{proof}
With Sagemath:
\begin{verbatim}
sage: D8 = DihedralGroup(4); D8.order()
8
sage: [r,s] = D8.gens(); r, s
((1,2,3,4), (1,4)(2,3))
sage: D8.centralizer(s).list()
[(), (1,3)(2,4), (1,4)(2,3), (1,2)(3,4)]
sage: D8.subgroup([s,r^2]).list()
[(), (1,3)(2,4), (1,4)(2,3), (1,2)(3,4)]
\end{verbatim}


$x$	$1$	$r$	$r^{2}$	$r^{3}$	$s$	$rs$	$r^{2} s$	$r^{3} s$

$C_{G} (x)$	$D_{8}$	$⟨ r ⟩$	$D_{8}$	$⟨ r ⟩$	$⟨ s, r^{2} ⟩$	$⟨ rs, r^{2} ⟩$	$⟨ s, r^{2} ⟩$	$⟨ rs, r^{2} ⟩$


$x$	$1$	$i$	$j$	$k$	$- 1$	$- i$	$- j$	$- k$

$C_{G} (x)$	$Q_{8}$	$⟨ i ⟩$	$⟨ j ⟩$	$⟨ k ⟩$	$Q_{8}$	$⟨ i ⟩$	$⟨ j ⟩$	$⟨ k ⟩$


$x$	$()$	$(1 2)$	$(1 3)$	$(2 3)$	$(1 2 3)$	$(1 3 2)$

$C_{G} (x)$	$S_{3}$	$⟨ (1 2) ⟩$	$⟨ (1 3) ⟩$	$⟨ (2 3) ⟩$	$⟨ (1 2 3) ⟩$	$⟨ (1 2 3) ⟩$


$x$	$1$	$r$	$r^{2}$	$r^{3}$	$r^{4}$	$r^{5}$	$r^{6}$	$r^{7}$

$C_{G} (x)$	$D_{16}$	$⟨ r ⟩$	$⟨ r ⟩$	$⟨ r ⟩$	$D_{16}$	$⟨ r ⟩$	$⟨ r ⟩$	$⟨ r ⟩$


$x$	$s$	$sr$	$s r^{2}$	$s r^{3}$	$s r^{4}$	$s r^{5}$	$s r^{6}$	$s r^{7}$

$C_{G} (x)$	$⟨ s, r^{4} ⟩$	$⟨ sr, r^{4} ⟩$	$⟨ s r^{2}, r^{4} ⟩$	$⟨ s r^{3}, r^{4} ⟩$	$⟨ s, r^{4} ⟩$	$⟨ sr, r^{4} ⟩$	$⟨ s r^{2}, r^{4} ⟩$	$⟨ s r^{3}, r^{4} ⟩$

Exercise 2.5.6 (Centralizers of every element in $D_8$, $Q_8$, $S_3$, $D_{16}$)

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Exercise 2.5.6 (Centralizers of every element in $D_8$, $Q_8$, $S_3$, $D_{16}$)

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