%%% Template for MA 361 homework, after Eric J. Malm's hmcpset class
%
%%% PREAMBLE -----------------------------------------------------------
% Do make any changes before \begin{document}, except to add macros
%
\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{boxedminipage}
\usepackage{graphicx}
\usepackage[margin=1in]{geometry}
\usepackage{mathpazo}
\usepackage{fancyhdr}

\usepackage{tikz}
\usetikzlibrary{arrows}

\setlength{\headheight}{15pt}
\setlength{\parindent}{0pt}

% include this to have lists enumerated by letters
\renewcommand{\labelenumi}{(\alph{enumi})}
\renewcommand{\labelenumii}{\roman{enumii}.}
\renewcommand{\labelenumiii}{\Alph{enumiii}.}



\newcommand{\collaborators}[1]{\begin{center}\large\sffamily{I collaborated with #1}\end{center}}

\newenvironment{problem}[1][]{%
  \bigskip% put space before problem statement box %
  \noindent\begin{boxedminipage}{\columnwidth}%
  \def\@tempa{#1}%
  \ifx\@tempa\empty\else%
    \textbf{#1}\hspace{0.5em}%
  \fi%
}{%
  \end{boxedminipage}%
}



\pagestyle{fancyplain}
\lhead{\textbf{\NAME}}
\chead{\textbf{\COURSE: HW\ \HWNUM}}
\rhead{\textbf{\DUEDATE}}
\cfoot{}


%%% MACROS -----------------------------------------------------------
% You may add more macros to this list
%
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}

\newcommand{\GL}{\operatorname{GL}}
\newcommand{\SL}{\operatorname{SL}}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\id}{\operatorname{id}}
\newcommand{\Inn}{\operatorname{Inn}}




















\begin{document}
%%% NAME AND ASSIGNMENT NUMBER -----------------------------------------------------------
% Edit the values below to identify yourself and the assignment
%

\newcommand\NAME{Name:}  % your name
\newcommand\HWNUM{16}  % the homework number
\newcommand\DUEDATE{11/11/2019} % the due date
\newcommand\COURSE{MA 361}  % the course number

%%% COLLABORATORS -----------------------------------------------------------------------------------
% If you worked with other students on this assignment, acknowledge them here
% If you did not collaborate with anyone, comment out this line
%

%\collaborators{Ferdinand Georg Frobenius and Emmy Noether}


%%% PROBLEMS AND SOLUTIONS -------------------------------------------------------------------
% Include all problems statements
% For solutions, use the \begin{proof} environment
% Note that \begin{proof}[Solution] will replace the label "Proof" with "Solution", etc.
%

\begin{problem}[1]
Let $G$ be a group acting on a set $X$. Let $\sim$ be the relation on $X$ defined by
\[ x_1 \sim x_2 \quad \text{if and only if} \quad \text{there exists $g \in G$ such that $gx_1 = x_2$.} \]
Prove that $\sim$ is an equivalence relation on $X$.
\end{problem}

%\begin{proof}
%
%\end{proof}


\newpage


\begin{problem}[2]
Let $G = \GL_2(\RR)$ be the group of invertible $2 \times 2$ matrices with entries from $\RR$ (with matrix multiplication as the group operation). Let
\[ X = \left\{
\begin{bmatrix}
	x \\
	y
\end{bmatrix}
\Bigm\vert
x, y \in \RR
\right\} \]
Define a function $G \times X \to X$ by matrix multiplication:
\[
\begin{bmatrix}
	a & b \\
	c & d \\
\end{bmatrix}
\cdot
\begin{bmatrix}
	x \\
	y
\end{bmatrix}
=
\begin{bmatrix}
	ax + by \\
	cx + dy
\end{bmatrix}.
\]

\begin{enumerate}
\item Prove that this is a group action.
\item Prove that this action is faithful.
\item Prove that this action is \emph{not} transitive.
\end{enumerate}
\end{problem}

%\begin{proof}
%
%\end{proof}


\newpage


\begin{problem}[3]
The group
\[ D_4 = \langle r, s \mid r^4 = s^2 = 1, rs = sr^{-1} \rangle \]
acts on the square below, where $r$ acts as a $90^{\circ}$ clockwise rotation and $s$ acts as a reflection across the axis of symmetry passing through the vertex labelled $a$.

\[
      \begin{tikzpicture}[scale = 4]
        \node at (0,0) {};
        \node at (0,1) {};
        \node at (1,0) {};
        \node at (1,1) {};
        \draw (0,0) -- (1,0);
        \draw (0,0) -- (0,1);
        \draw (1,0) -- (1,1);
        \draw (0,1) -- (1,1);
        \node at (0,0) [label = below : \(g\)] {\textbullet};
        \node at (0,0.5) [label = left : \(d\)] {\textbullet};
        \node at (0,1) [label = above : \(a\)] {\textbullet};
        \node at (0.5,0) [label = below : \(h\)] {\textbullet};
        \node at (0.5,0.5) [label = left : \(e\)] {\textbullet};
        \node at (0.5,1) [label = above : \(b\)] {\textbullet};
        \node at (1,0) [label = below : \(i\)] {\textbullet};
        \node at (1,0.5) [label = right : \(f\)] {\textbullet};
        \node at (1,1) [label = above : \(c\)] {\textbullet};
      \end{tikzpicture}
\]

We consider the action of $D_4$ on the set $X = \{a, b, c, \ldots, i\}$ of nine labelled points.

    \begin{enumerate}
    \item Find all of the orbits of the action of $D_4$ on $X$.
    \item Find the \textbf{stabilizer subgroup} $G_a = \{g \in D_4 \mid g \cdot a = a\}$ of the point $a$.
    \item Find the stabilizer subgroup $G_b$ of the point $b$.
    \item Find the \textbf{fixed points} of the element $r$. That is, find all $x \in X$ such that $r \cdot x = x$.
    \item Find the fixed points of the element $sr$.
    \end{enumerate}
\end{problem}

%\begin{proof}
%
%\end{proof}



\end{document}
