Table of Contents

Automorphisms Of A Directed Square

Problem

Consider the directed graph shown below with vertex set and arrows (directed edges)

%center%http://bmw.byuimath.com/aa/lib/exe/fetch.php?media=problem:digraphsquare.png%%

  1. How would you define an automorphism of a directed graph?
  2. List all the automorphisms of this directed graph. You should have 4.
  3. Is there is an automorphism such that (the composition) is the identity automorphism, but is not the identity?
  4. Is there is an such that is the identity but is not the identity?

# Is there is an such that every other is of the form for some ?

  1. For every , do we have ?

Remarks

$\LaTeX$ version

problem.automorphisms_of_a_directed_square.tex
%%%%%
% DEPENDENCIES
% RequiredPackages: \usepackage{tikz}
% RequiredMacros: \DeclareMathOperator{\aut}{Aut} 
%%%%%
\begin{problem}
Consider the directed graph $\mathcal{G} = (V,A)$ shown below with vertex set $V = \{1,2,3,4\}$ and arrows (directed edges) $$A = \{(1,2),(2,3),(3,4),(4,1)\}.$$
 
%center%http://bmw.byuimath.com/aa/lib/exe/fetch.php?media=problem:digraphsquare.png%%
 
\begin{enumerate}
\item How would you define an automorphism of a directed graph?
\item List all the automorphisms of this directed graph. You should have 4. 
\item Is 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 Is 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 Is 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 $a,b\in \aut(\mathcal{G})$, do we have $a\circ b = b\circ a$?
\end{problem}