The goal of this problem is to find a way to represent the symmetric group with diagrams. We will focus on .

1. Which element of does the following diagram seem to represent?
2. What is the diagram for the inverse of the previous element.
3. Formulate a diagrammatic rule for finding the diagram of the inverse of an element of .
4. What is the diagram for the identity.
5. Consider whose diagrams are given below. Determine the diagrams for and .
6. Formulate a diagrammatic rule for finding the diagram of the composition of two elements of .
7. Find a such that is not the identity but is the identity. Find more (different) elements in with the same property.
8. Find a such that is not the identity but is the identity.

• Answers to the problem, of course, depend on whether a diagram is read top-to-bottom or bottom-to-top and whether function composition is performed left-to-right or right-to-left.

problem.diagramatic_representation_of_sn.tex

%%%%%
% DEPENDENCIES
% RequiredPackages \usepackage{tikz}
%%%%%
\begin{problem}
The goal of this problem is to find a way to represent the symmetric group $S_n$ with diagrams. We will focus on $S_4$.
\begin{enumerate}
\item Which element of $S_4$ does the following diagram seem to represent?
\[\begin{tikzpicture}[vertex/.style={circle,draw=black,fill=blue!15,thick,inner sep=1.5pt}, line width = 2]
\foreach \i in {1,2,3,4} {
\draw (\i,2) node[vertex] (top\i) {\footnotesize$\i$};
\draw (\i,0) node[vertex] (bottom\i) {\footnotesize$\i$};
}
\draw (top1) -- (bottom1);
\draw (top2) -- (bottom3);
\draw (top3) -- (bottom4);
\draw (top4) -- (bottom2);
\end{tikzpicture}\]
\item What is the diagram for the inverse of the previous element.
\item Formulate a diagrammatic rule for finding the diagram of the inverse of an element of $S_4$.
\item What is the diagram for the identity.
\item Consider $\sigma,\tau\in S_4$ whose diagrams are given below. Determine the diagrams for $\sigma\tau$ and $\sigma\tau$.
\[\sigma = \begin{tikzpicture}[baseline= (current bounding box.west),vertex/.style={circle,draw=black,fill=blue!15,thick,inner sep=1.5pt}, line width = 2]
\foreach \i in {1,2,3,4} {
\draw (\i,2) node[vertex] (top\i) {\footnotesize$\i$};
\draw (\i,0) node[vertex] (bottom\i) {\footnotesize$\i$};
}
\draw (top1) -- (bottom1);
\draw (top2) -- (bottom3);
\draw (top3) -- (bottom4);
\draw (top4) -- (bottom2);
\end{tikzpicture}\quad\quad
\tau = \begin{tikzpicture}[baseline= (current bounding box.west),vertex/.style={circle,draw=black,fill=blue!15,thick,inner sep=1.5pt}, line width = 2]
\foreach \i in {1,2,3,4} {
\draw (\i,2) node[vertex] (top\i) {\footnotesize$\i$};
\draw (\i,0) node[vertex] (bottom\i) {\footnotesize$\i$};
}
\draw (top1) -- (bottom2);
\draw (top2) -- (bottom1);
\draw (top3) -- (bottom4);
\draw (top4) -- (bottom3);
\end{tikzpicture}\]
\item Formulate a diagrammatic rule for finding the diagram of the composition of two elements of $S_4$.
\item Find a $\sigma \in S_4$ such that $\sigma$ is not the identity but $\sigma^2$ is the identity. Find $5$ more (different) elements in $S_4$ with the same property.
\item Find a $\sigma \in S_4$ such that $\sigma$ is not the identity but $\sigma^3$ is the identity.
\end{enumerate}
\end{problem}

• Symmetric group