====== Automorphisms Of A Directed Square ======
==== 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%%

  - How would you define an automorphism of a directed graph?
  - List all the automorphisms of this directed graph. You should have 4. 
  - 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?
  - 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?
# 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 $n$?
  - For every $a,b\in \aut(\mathcal{G})$, do we have $a\circ b = b\circ a$?

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==== Remarks ====  
  * Make remarks with a list.

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==== $\LaTeX$ version ====
<file tex 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}
</file>

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==== External links ====
  * [[wp>Dihedral group]]

{{tag>problem ben}}