\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{\inv}{^{-1}} % inverse
\newcommand{\into}{\hookrightarrow}  % injection
\newcommand{\onto}{\twoheadrightarrow} % surjection
\newcommand{\map}[1]{\xrightarrow{#1}} % named map
\newcommand{\id}{\mathrm{id}} % identity map
\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 HW02}
\author{[ADD YOUR NAME HERE]}
\date{Due Tuesday, 10/17/23 at 11:59 pm on Gradescope}

\begin{document}

\maketitle

\begin{reminder} 
Your homework submission must be typed 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}
Parts of Chp.\ 2 and 3, esp.\ \S 2.2--2.5, 3.2--3.3, 11.1.
Groups, abelian groups, order of a group or element of a group, cyclic groups, subgroups; definition of rings, fields, and vector spaces; homomorphisms, normal subgroups, 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. I will reveal which problems are fully graded in the solutions, which will be posted on the Friday following the due date.

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

Let $\ord(g) \in \N \cup \{\infty\}$ denote the order of an element $g$ in a group $G$. 
Recall that $\ord(g) = | \langle g \rangle |$ or, equivalently, 
\[
    \ord(g) = \min \left \{ n\in \N \st g^n = 1 \right \}.
\]
Let $a,b \in G$. Prove that $\ord(ab) = \ord(ba)$. 

\solution


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

Show by example that the product of elements of finite order in a group need not have finite order. 
What if the group is abelian?

\solution


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

Prove that every subgroup of a cyclic group is cyclic. 
\textit{Hint: Work with exponents and use the description of the subgroups of $\Z^+$.}

\solution





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

Let $U$ denote the group of invertible upper triangular $2 \times 2$  matrices 
\[
    \left \{
        \begin{bmatrix}
            a & b \\ 0 & d 
        \end{bmatrix} 
        \st a,b,d \in \R, \ ad\neq 0
    \right \} \subset GL_n(\R)
\]
and let $\varphi: U \to \R^\times$ be the map that sends $A \mapsto a^2$. Prove that $\varphi$ is a homomorphism, and determine its kernel and image.


\solution

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
Let $f: \R^+ \to \C^\times$ be the map $f(x) = e^{ix}$. Prove that $f$ is a homomorphism, and determine its kernel and image.

\solution






\end{document}
