\documentclass[12pt,std]{article}
\usepackage{amsmath,amsthm,amsfonts,amssymb,latexsym, hyperref}

\begin{document}

\begin{center} \textbf{Square Symmetries Multiplication Table.} \end{center}

\

The set of ``symmetries'' of the square is listed below:

\

\begin{eqnarray*}
I:  \left [ \begin{array}{cc}
1 & 2  \\ 4 & 3
\end{array} \right ] \rightarrow  \left [ \begin{array}{cc}
1 & 2  \\ 4 & 3
\end{array} \right ] &  & R:  \left [ \begin{array}{cc}
1 & 2  \\ 4 & 3
\end{array} \right ] \rightarrow  \left [ \begin{array}{cc}
4 & 1  \\ 3 & 2
\end{array} \right ] \\ \\
R^2:  \left [ \begin{array}{cc}
1 & 2  \\ 4 & 3
\end{array} \right ] \rightarrow  \left [ \begin{array}{cc}
3 & 4  \\ 2 & 1
\end{array} \right ] &  & R^3:  \left [ \begin{array}{cc}
1 & 2  \\ 4 & 3
\end{array} \right ] \rightarrow  \left [ \begin{array}{cc}
2 & 3  \\ 1 & 4
\end{array} \right ] \\
\\
V:  \left [ \begin{array}{cc}
1 & 2  \\ 4 & 3
\end{array} \right ] \rightarrow  \left [ \begin{array}{cc}
2 & 1  \\ 3 & 4
\end{array} \right ] &  & H:  \left [ \begin{array}{cc}
1 & 2  \\ 4 & 3
\end{array} \right ] \rightarrow  \left [ \begin{array}{cc}
4 & 3  \\ 1 & 2
\end{array} \right ] \\ \\
U:  \left [ \begin{array}{cc}
1 & 2  \\ 4 & 3
\end{array} \right ] \rightarrow  \left [ \begin{array}{cc}
3 & 2 \\ 4 & 1
\end{array} \right ] &  & L:  \left [ \begin{array}{cc}
1 & 2  \\ 4 & 3
\end{array} \right ] \rightarrow  \left [ \begin{array}{cc}
1 & 4  \\ 2 & 3
\end{array} \right ] \\
\end{eqnarray*}

\

To multiply two of them together you operate with one and then the other.  You should convince yourself that this operation is not commutative; so it make a difference the order in which the operations are done.  I'll base my notation on the way we are used to thinking about functions. So $R$ times $V$ means you operate first by $V$ and then by $R$; in other words, as follows:

$$RV \left ( \left [ \begin{array}{cc}
1 & 2  \\ 4 & 3
\end{array} \right ] \right ) = R \Big( V \left ( \left [ \begin{array}{cc}
1 & 2  \\ 4 & 3
\end{array} \right ] \right ) \Big) = R  \left ( \left [ \begin{array}{cc}
2 & 1  \\ 3 & 4
\end{array} \right ] \right ) = \left [ \begin{array}{cc}
3 & 2  \\ 4 & 1
\end{array} \right ]. $$
So $RV = U$.

\

Exercise 1.  Use this information to construct the multiplication table for the symmetries of the square.  [Hint: it's a group; so you can use the group properties to fill out the table more quickly (e.g it has an identity and inverses and it's a sudoku solution).]

\

Exercise 2. Show that the triangle symmetries is isomorphic to $S_3$.





\end{document}