Table of Contents

Automorphism group of a square

Problem

Consider the graph drawn below. The vertex set is and the (symmetric) relation giving adjacency is . Specifically, and Write down all elements of ; view these as the elements of that preserve . Also, determine if the following statements are true or false; explain your answers. For , denotes the (function) composition of with , and denotes the composition of with itself -times.

  1. There is an such that is the identity but is not the identity.
  2. There is an such that is the identity but is not the identity.
  3. There is an such that every other is of the form for some .
  4. For every , we have .

Remarks

$\LaTeX$ version

problem.aut-square.tex
%%%%%
% DEPENDENCIES
% RequiredPackages \usepackage{tikz}
% RequiredMacros: \DeclareMathOperator{\aut}{Aut}  
%%%%%
\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. For $a,b\in \aut(\mathcal{G})$, $ab$ denotes the (function) composition of $a$ with $b$, and $a^n$ denotes the composition of $a$ with itself $n$-times.
\begin{enumerate}
\item There is an $a\in \aut(\mathcal{G})$ such that $a^2$ is the identity but $a$ is not the identity.
\item There is an $a\in \aut(\mathcal{G})$ such that $a^3$ is the identity but $a$ is not the identity.
\item There is an $a\in \aut(\mathcal{G})$ 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}