\documentclass[11pt]{article}
\usepackage[utf8]{inputenc}

\usepackage[margin=1in]{geometry}
\usepackage{enumitem}
\usepackage{amsmath, amssymb, amsthm}
\usepackage{tikz, tikz-cd}
\usetikzlibrary{decorations.markings}
\usepackage{xcolor}
\usepackage{verbatim}
\usepackage{graphicx}
\usepackage{hyperref}
\hypersetup{
    colorlinks=true,
    linkcolor=blue,
    filecolor=magenta,      
    urlcolor=blue,
    pdftitle={Knot Theory},
    pdfpagemode=FullScreen,
    }
% \usepackage{ulem}

% THEOREM ENVIRONMENTS
\theoremstyle{definition}
\newtheorem*{theorem}{{Theorem}}
\newtheorem*{lemma}{{Lemma}}
\newtheorem*{proposition}{{Proposition}}
\newtheorem*{definition}{{Definition}}
\newtheorem*{corollary}{{Corollary}}
\newtheorem*{example}{{Example}}
\newtheorem*{remark}{{Remark}}
\newtheorem*{claim}{{Claim}}
\newtheorem*{question}{{Question}}

\theoremstyle{plain}
\newtheorem*{reminder}{{\textit{Reminder}}}
\newtheorem*{hint}{{\textit{Hint}}}



% COMMANDS
\newcommand{\solution}{{\color{blue} \textsc{\ \\ Solution.\ \ }}}
\newcommand{\Z}{\mathbb{Z}} % the integers
\newcommand{\N}{\mathbb{N}} % natural numbers
\newcommand{\R}{\mathbb{R}} % real numbers
\newcommand{\C}{\mathbb{C}} % complex numbers
\newcommand{\Q}{\mathbb{Q}} % rational numbers
% \newcommand{\st}{ \ : \ } % such that
\newcommand{\st}{ \ | \ } % such that
\newcommand{\inv}{^{-1}} % inverse
\newcommand{\into}{\hookrightarrow}  % injection
\newcommand{\onto}{\twoheadrightarrow} % surjection
\newcommand{\map}[1]{\xrightarrow{#1}} % named map
\newcommand{\id}{\mathrm{id}} % identity map
\newcommand{\Frac}{\mathrm{Frac}} % fraction field
\newcommand{\normal}{\unlhd} % normal subgroup of
\newcommand{\F}{\mathbb{F}} % random field
\newcommand{\Isom}{\mathrm{Isom}} % isometries group

\DeclareMathOperator{\img}{im} % image
\DeclareMathOperator{\sgn}{sgn} % sign of permutation
\DeclareMathOperator{\ord}{ord} % order of group element
\DeclareMathOperator{\lcm}{lcm} % least common multiple
% \DeclareMathOperator{\gcd}{gcd} % greatest common divisor

% \newcommand{\bpmat}{\begin{pmatrix}}
% \newcommand{\epmat}{\end{pmatrix}}


% knot theory
\newcommand{\lk}{\mathrm{lk}} % linking number


% SOME ENVIRONMENTS

\newcommand{\exheading}[1]{\section*{Exercise #1}}
\newcounter{exnum}
% \setcounter{exnum}{1} % default 0 start
\newcommand{\exercise}{
    \stepcounter{exnum} 
    \exheading{\theexnum}
    }

\newcommand{\bea}{\begin{enumerate}[label={(\alph*)}]}
\newcommand{\ee}{\end{enumerate}}
\newcommand{\bmat}{\begin{bmatrix}}
\newcommand{\emat}{\end{bmatrix}}






\title{MAT 150A HW08}
\author{[add your name here]}
\date{Due Tuesday, 12/5/23 at 11:59 pm on Gradescope}

\begin{document}

\maketitle

\begin{reminder} 
Your homework submission \textbf{\large must be typed} (TeX'ed) up in full sentences, with proper mathematical formatting. 

\begin{comment}
The following resources may be useful as you learn to use TeX and Overleaf:
\begin{itemize}
    \item Overleaf's introduction to LaTeX: \\ \url{http://www.overleaf.com/learn/latex/Learn_LaTeX_in_30_minutes}
    \item Detexify: \\ \url{http://detexify.kirelabs.org/classify.html}
\end{itemize}
\end{comment}
\end{reminder}


\paragraph{Covered in this HW}
\S 6.7, 6.8, 6.9: group actions (i.e.\ group operations), orbits, stabilizers; the orbit-stabilizer theorem, counting formula


\paragraph{Grading} 
Some of the (parts of) problems will be graded in detail out of several points, and necessary feedback will be given. The rest will be graded out of 2 points. 




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise

Does the rule $P * A = PAP^\top$ define an operation of $GL_n$ on $M_{n\times n}$, the set of $n \times n$ matrices? 
\textit{Here, $P^\top$ is the transpose of the matrix $P \in GL_n$.}


\solution


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
What is the stabilizer of the coset $[aH]$ for the operation fo $G$ on $G/H$?

\solution

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise

Exhibit the bijective map $\varepsilon$ from the orbit-stabilizer theorem explicitly, for the case where $G$ is the dihedral group $D_4$ and $S$ is the set of vertices of a square.

\solution

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise

A \emph{cube} is a 3D solid with 6 square faces of equal size:
\begin{center}
\begin{tikzpicture}[scale=3]
    \def \s{0.5};
    \def \t{0.4};
    \draw (0+\s, 0+\t) rectangle (1+\s, 1+\t);
    \draw[line width=.3cm, white] (0,0) rectangle (1,1);
    \draw (0,0) rectangle (1,1);
    \foreach \x in {0,1}{
        \foreach \y in {0,1}{
            \draw (\x, \y) -- (\x+\s,\y+\t);
        }
    }
\end{tikzpicture}
\end{center}
One example of the cube is the set of points $Q = [0,1]^3 \subset \R^3$. 

Let $G$ be the group of \textbf{rotational symmetries} of the cube. This is a subgroup of $O(3)$ consisting of \emph{orientation-preserving} symmetries of the cube. \footnote{The group of orientation-preserving isometries of $\R^3$ is called $SO(3)$.}

Let $V$, $E$, and $F$ denote the sets of vertices, edges, and faces of the cube, respectively. Check for yourself that the size of these sets are 
\[
    |V| = 8 \quad |E| = 12 \quad |F| = 6.
\]


\bea

\item Use the counting formula to determine the order of $G$.

\item Let $G_v, G_e, G_f$ be the stabilizers of a vertex $v$, and edge $e$, and a face $f$ of the cube.
Determine the formulas of the form 
\[
    |S| = |O_1| + |O_2| + \cdots + |O_k|
\]
(formula 6.9.4 in the text)
that represent the decomposition of each of the three sets $V, E, F$ into orbits for each of the subgroups. 
\emph{Your solution should contain $9=3 \times 3$ formulas, one for each (group, set) pair. First make sure you are clear on what the group and set in the group action is, in each case!}

\ee


\solution



\end{document}
