\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

\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}}




\title{MAT 150A HW05}
\author{[add your name here]}
\date{Due Tuesday, 11/7/23 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. The following resources may be useful as you learn to use TeX and Overleaf:
\begin{itemize}
    \item Overleaf's introduction to LaTeX: \\ \url{https://www.overleaf.com/learn/latex/Learn_LaTeX_in_30_minutes}
    \item Detexify: \\ \url{https://detexify.kirelabs.org/classify.html}
\end{itemize}
\end{reminder}

\paragraph{Covered in this HW}
\S 2.10, 2.12; 3.2; parts of 7.9, 7.10. Correspondence theorem, quotient groups, first isomorphism theorem, generators and relations.


\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. 


%----
\begin{remark}
    Mathematicians sometimes use $*$ to indicate ``some value''. For example, below, the matrix on the left represents the entire set of $2 \times 2$ upper triangular matrices over a field $\F$, on the right:
    \[
        \begin{bmatrix}
            * & * \\
            0 & *
        \end{bmatrix}
        = 
        \left \{
        \begin{bmatrix}
            a & b \\ 
            0 & c 
        \end{bmatrix}
        \st 
        a,b,c \in \F
        \right \}.
    \]
\end{remark}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
Let $G$ be a group, let $N \normal G$, and let $\bar G = G/N$.
Prove that if $G$ is generated $x,y \in G$, then $\bar G$ is generated by $\bar x, \bar y \in \bar G$. 

\solution


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
% Heisenberg group
In the general linear group $GL_3(\F)$, consider the subsets 
\[
    H = 
    \begin{bmatrix}
        1 & * & * \\
        0 & 1 & *  \\
        0 & 0  & 1 
    \end{bmatrix}
    \qquad
    \text{and}
    \qquad
    K = 
    \begin{bmatrix}
        1 & 0 & * \\
        0 & 1 & 0  \\
        0 & 0  & 1 
    \end{bmatrix}
\]
where $*$ represents an arbitrary element of a field $\F$. 


\bea
\item Show that $H$ is a subgroup of $GL_3(\F)$. \emph{Hint: First, compute the product 
\[
    \begin{bmatrix}
        1 & a & b \\
        0 & 1 & c \\
        0 & 0 & 1 
    \end{bmatrix}
    \begin{bmatrix}
        1 & x & y \\
        0 & 1 & z \\
        0 & 0 & 1 
    \end{bmatrix}.
\]
}
\item Show that $K$ is a \emph{normal} subgroup of $H$.
\item For $\F = \R$, identify the quotient group $H/K$ (up to isomorphism).
\emph{Hint: Let $A,B \in H$. Under what conditions are $A$ and $B$ in the same coset of $K$? Use this to construct a surjective homomorphism from $H$.}
\ee

\begin{remark} 
The subgroup $H$ discussed here is called the \emph{Heisenberg group}, and we can actually define it using elements of commutative rings, not just fields. This version of this group with $\F = \R$ was used by Weyl to give an algebraic interpretation of Heisenberg's Uncertainty Principle.
\end{remark}


\solution





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise 
% S_4 example with conjugation classes
Recall that the Klein four group is $V = \{1, a, b, ab\} = \langle a,b \st a^2 = b^2 = [a,b] = 1\rangle \cong C_2 \times C_2$ (see page 47 in the book).

\bea

\item Prove that the subgroup $N = \{ e, (1~2)(3~4), (1~3)(2~4), (1~4)(2~3)\}$ in $S_4$ is isomorphic to the Klein four group. 

\item Prove that $N$ is normal in $S_4$. \emph{Hint: Use a theorem from lecture on Wednesday; do not use brute force!}

\item Prove that the subgroup $H = \langle (1~2) , (3~4) \rangle  \leq S_4$ is also isomorphic to $V$, but is not a normal subgroup of $S_4$. 

\item Identify the quotient group $S_4/N$ by computing the cosets. \emph{Hint: Recall that $|S_4| = 4! = 24$; use the counting formula. Either define an isomorphism between $S_4/N$ and your candidate group, or define a surjection from $S_4$ to your candidate group. You do not need to show me that your map is a homomorphism; just check for yourself that it really is.}

\textbf{Inserting images:} When computing the cosets, you may write these by hand and insert a clear image of your work, using 
\begin{verbatim}
    \includegraphics[width=\textwidth]{your-image-name.png}
\end{verbatim}

\item How many subgroups are there in $S_4$ that contain $N$?  (\emph{Do not solve this by brute force!})




\ee


\solution



\end{document}
