Table of Contents

Automorphismsofasquare2

Problem

Again consider the graph $\mathcal{G} = (V,E)$ shown below with vertex set $V = \{1,2,3,4\}$ and edges $$E = \{\{1,2\},\{2,3\},\{3,4\},\{1,4\}\}.$$

%center%http://bmw.byuimath.com/aa/lib/exe/fetch.php?cache=&media=wiki:graphpictures:labeledsquare.png%%

Recall that set of all automorphisms of $\mathcal{G}$ is written $\aut(\mathcal{G})$. We listed all the elements in this set in problem Automorphisms Of A Square. Determine if each statement below is true or false. Make sure you justify your answers.

  1. There is an automorphism $a\in \aut(\mathcal{G})$ such that $a^2=a\circ a$ (the composition) is the identity automorphism, but $a$ is not the identity.
  2. There is an $a\in \aut(\mathcal{G})$ such that $a^3=a\circ a\circ a$ is the identity but $a$ is not the identity.
  3. 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 $n$. [Hint: for each element in $\aut(G)$, find the smallest $n$ such that $a^n=id_X$. ]
  4. For every $a,b\in \aut(\mathcal{G})$, we have $a\circ b = b\circ a$.

Remarks

$\LaTeX$ version

problem.automorphismsofasquare2.tex
%%%%%
% DEPENDENCIES
% RequiredPackages: \usepackage{tikz}
% RequiredMacros: \DeclareMathOperator{\aut}{Aut} 
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\begin{problem}
Again consider the graph $\mathcal{G} = (V,E)$ shown below with vertex set $V = \{1,2,3,4\}$ and edges $$E = \{\{1,2\},\{2,3\},\{3,4\},\{1,4\}\}.$$
 
%center%http://bmw.byuimath.com/aa/lib/exe/fetch.php?cache=&media=wiki:graphpictures:labeledsquare.png%%
 
Recall that set of all automorphisms of $\mathcal{G}$ is written $\aut(\mathcal{G})$. We listed all the elements in this set in problem [[Problem/AutomorphismsOfASquare]].
\begin{enumerate}
Determine if each statement below is true or false. Make sure you justify your answers.  
\item There is an automorphism $a\in \aut(\mathcal{G})$ such that $a^2=a\circ a$ (the composition) is the identity automorphism, but $a$ is not the identity.
\item There is an $a\in \aut(\mathcal{G})$ such that $a^3=a\circ a\circ a$ 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 $n$. [Hint: for each element in $\aut(G)$, find the smallest $n$ such that $a^n=id_X$. ]
\item For every $a,b\in \aut(\mathcal{G})$, we have $a\circ b = b\circ a$.
\end{problem}

problem ben