\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={Algebra},
pdfpagemode=FullScreen,
}
% 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{\acton}{\curvearrowright} % group action
\newcommand{\floor}[1]{\lfloor #1 \rfloor} % floor function
% SOME ENVIRONMENTS
\newcommand{\exheading}[1]{\section*{Exercise #1}}
\newcounter{exnum}
% \setcounter{exnum}{1} % default 0 start
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}
\newcommand{\bea}{\begin{enumerate}[label={(\alph*)}]}
\newcommand{\ee}{\end{enumerate}}
% round matrix
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% square matrix
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% TIKZ
\newcommand{\eqtriangle}{\draw (0,0) -- (1,0) -- (.5, .866) -- (0,0)} % equilateral triangle in tikz
\title{MAT 150A HW09}
\author{[add your name here]}
\date{Due Tuesday, 3/12/24 at 11:59 pm on Gradescope}
\begin{document}
\maketitle
{
\scriptsize
\paragraph{Proof-based course}
This is a proof-based course and you are expected to \textbf{clearly prove} all your claims. If you're wondering how much detail to include, a good rule of thumb is that your proofs should be slightly more detailed than the proofs in the book, but not less detailed. They should also not be unreasonably verbose.
\paragraph{Reminder}
Homeworks must be typed using LaTeX \textbf{in full sentences with proper mathematical formatting}. Handwritten homeworks will not be accepted. If there is a documented reason why you can't type up your homework, let me know and we can discuss an alternate policy. Otherwise, please consider learning how to properly write and typeset mathematics as part of this course.
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 $P \in GL_n$.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
Suppose a group $G$ \textbf{acts freely} on a set $S$ (i.e.\ the group action $G \acton S$ is free). Prove that for any $s \in S$, the stabilizer $G_s$ is the trivial subgroup of $G$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
What is the stabilizer of the coset $[aH]$ for the action of $G$ on $G/H$?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
Let $G = GL_n(\R)$ act on the set $V = \R^n$ by left multiplication.
\bea
\item Describe the decomposition of $V$ into orbits for this action.
\item What is the stabilizer of $e_1$?
\item Is this action of $G$ on $V - \{0\}$ free, transitive, both, or neither?
\ee
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
Let $G$ be the group of rotational symmetries of a cube. Let $V$, $E$, and $F$ denote the set of vertices, edges, and faces of a cube, respectively. Check for yourself that the size of these sets are
\[
|V| = 8 \quad |E| = 12 \quad |F| = 6.
\]
Fix a vertex $v \in V$, an edge $e \in E$, and a face $f \in F$, and let $G_v$, $G_e$, and $G_f$ be their stabilizers, respectively.
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 under the action of each of the subgroups of $G$.
\paragraph{Note} Your solution should contain $9=3 \times 3$ formulas, one for each (group, set) pair, such as $G_v \acton V$, $G_v \acton E$, etc. You should explain any geometric reasoning in words; if you'd like, you can also include a picture by following the instructions \href{https://www.overleaf.com/learn/how-to/Including_images_on_Overleaf}{here}. The tl;dr is that you should use the following code:
\begin{verbatim}
\includegraphics[width=4cm]{yourImageName}
\end{verbatim}
\end{document}