Starting Assignment 3

MIT/Problems/18.703/Assignment3.tex

@@ -0,0 +1,22 @@
+\pagebreak
+\section*{Problem Set 3}
+\hypertarget{ModernAlgebra-Assignment03}{}
+
+\begin{enumerate}
+    \item{Ex 8: 44, 45, 47}
+    \item{Ex 9: 7, 11, 13abc, 15, 23}
+\end{enumerate}
+
+Bonus problems:
+\begin{enumerate}
+    \item{
+        Prove the following:
+        \begin{enumerate}
+            \item{The group of orientation preserving symmetries of a regular tetrahedron is isomorphic to $ A_4 $ and of a cube to $ S_4 $.}
+            \item{The group of all symmetries of a tetrahedron is isomorphic to $ S_4 $.}
+            \item{A harder problem: the group of orientation preserving symmetries of a regular icosahedron is isomorphic to $ A_5 $.}
+        \end{enumerate}
+    }
+    \item{Prove that all finite groups $ G $ of symmetries of a plane are isomorphic either to $ C_n $ for $ n \ge 1 $ or to $ D_n $. (Hint: Prove that $ G $ has fixed point.)}
+    \item{Prove that the Braid group $ B_n $ is a group.}
+\end{enumerate}

MIT/Solutions/18.703/Assignment2/Ex9.tex

@@ -1,4 +1,4 @@
-\includepdf[pages=15]{MIT/Problems/18.703/fraleigh-excerpts.pdf}
+\includepdf[pages=16]{MIT/Problems/18.703/fraleigh-excerpts.pdf}
 
 \input{MIT/Solutions/18.703/Assignment2/Ex9/1.tex}
 \input{MIT/Solutions/18.703/Assignment2/Ex9/10.tex}

MIT/Solutions/18.703/Assignment3/Ex8.tex

@@ -0,0 +1,5 @@
+\includepdf[pages=15]{MIT/Problems/18.703/fraleigh-excerpts.pdf}
+
+\input{MIT/Solutions/18.703/Assignment3/Ex8/44.tex}
+\input{MIT/Solutions/18.703/Assignment3/Ex8/45.tex}
+\input{MIT/Solutions/18.703/Assignment3/Ex8/47.tex}

MIT/Solutions/18.703/Assignment3/Ex8/44.tex

@@ -0,0 +1,4 @@
+\section*{Ex8 - 44}
+One can either rotate this polygon or reflece this polygon. Rotation generates a subgroup of order $ n $. Reflecting once generate another unique configuration, whereas we can rotate it $ n $ times to generate $ n $ more configurations, so the group has $ 2n $ elements.
+
+TODO: Is these description precise enough? How do I write this better?

MIT/Solutions/18.703/Assignment3/Ex8/45.tex

@@ -0,0 +1,6 @@
+\section*{Ex8 - 45}
+A vertex of a cube is always connected to 3 other vertices. If we fix a vertex, we can choose any of the 3 other vertices to be the one at its left. That way we produces 3 configuration per vertex, a total of 24 configurations.
+
+The rotation along the X, Y, Z axis for 90 degree generates 3 subgroups of order 4.
+
+The rotation along the diagonal generates a subgroup of order 3. There are 4 such diagonals.

MIT/Solutions/18.703/Assignment3/Ex8/47.tex

@@ -0,0 +1,2 @@
+\section*{Ex8 - 47}
+TODO

MIT/Solutions/MIT.tex

@@ -7,7 +7,7 @@
 \usepackage{mathrsfs}
 
 \newcommand{\emptymacro}{}    % Used to test against an empty macro
-\newcommand{\linearalgebra}{} 
+% \newcommand{\linearalgebra}{} 
 
 \title{MIT Math Problems}
 \author{Cecilia}
@@ -37,6 +37,7 @@ \section*{Links}
         \begin{enumerate}
             \item \hyperlink{ModernAlgebra-Assignment01}{Assignment1}
             \item \hyperlink{ModernAlgebra-Assignment02}{Assignment2}
+            \item \hyperlink{ModernAlgebra-Assignment03}{Assignment3}
         \end{enumerate}
     \end{enumerate}
 
@@ -126,4 +127,6 @@ \section*{Links}
 \input{MIT/Solutions/18.703/Assignment2/Ex8.tex}
 \input{MIT/Solutions/18.703/Assignment2/Ex9.tex}
 \input{MIT/Solutions/18.703/Assignment2/Bonus.tex}
+\input{MIT/Problems/18.703/Assignment3.tex}
+\input{MIT/Solutions/18.703/Assignment3/Ex8.tex}
 \end{document}