Problem. Consider the graph $\mathcal{G} = (V,E)$ drawn below. The vertex set is $V$ and the (symmetric) relation giving adjacency is $E$. Specifically, $V = \{1,2,3,4\}$, and \[E = \{(1,2),(2,1),(2,3),(3,2),(3,4),(4,3),(1,4),(4,1)\}.\] Write down all elements of $\aut(\mathcal{G})$; view these as the elements of $S_4$ that preserve $E$. Also, determine if the following statements are true or false; explain your answers.

1. There some $a\in \aut(\mathcal{G})$ such that $a^2$ is the identity.
2. There some $a\in \aut(\mathcal{G})$ such that $a^3$ is the identity.
3. There some $a\in \aut(\mathcal{G})$ such that every other $b\in \aut(\mathcal{G})$ is of the form $b=a^n$ for some $n$.
4. For every $a,b\in \aut(\mathcal{G})$, $ab = ba$.

• $\aut(\mathcal{G})$ denotes the automorphism group of $\mathcal{G}$. The definition of an automorphism group also explains the notation $a^n$ and $ab$.
• $S_4$ denotes the symmetric group on $\{1,2,3,4\}$.

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% DEPENDENCIES
% RequiredPackages: \usepackage{tikz}
% RequiredMacros: \DeclareMathOperator{\aut}{Aut} 
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\begin{problem}
Consider the graph $\mathcal{G} = (V,E)$ drawn below. The vertex set is $V$ and the (symmetric) relation giving adjacency is $E$. Specifically, $V = \{1,2,3,4\}$, and \[E = \{(1,2),(2,1),(2,3),(3,2),(3,4),(4,3),(1,4),(4,1)\}.\]
Write down all elements of $\aut(\mathcal{G})$; view these as the elements of $S_4$ that preserve $E$. Also, determine if the following statements are true or false; explain your answers.
\begin{enumerate}
\item There some $a\in \aut(\mathcal{G})$ such that $a^2$ is the identity.
\item There some $a\in \aut(\mathcal{G})$ such that $a^3$ is the identity.
\item There some $a\in \aut(\mathcal{G})$ such that every other $b\in  \aut(\mathcal{G})$ is of the form $b=a^n$ for some $n$. 
\item For every $a,b\in \aut(\mathcal{G})$, $ab = ba$. 
\end{enumerate}
\begin{center}
\begin{tikzpicture}[vertex/.style={circle,draw=black,fill=blue!15,thick,inner sep=1.5pt}, line width = 2]
\draw (0,0) node[vertex] {\footnotesize$1$} -- (0,1) node[vertex] {\footnotesize$2$} -- (1,1) node[vertex] {\footnotesize$3$} -- (1,0) node[vertex] {\footnotesize$4$} -- cycle;
\end{tikzpicture}
\end{center}
\end{problem}

• Dihedral group