\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,
    }

% 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{\floor}[1]{\lfloor #1 \rfloor} % floor function

% 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}}
% round matrix
\newcommand{\bpmat}{\begin{pmatrix}}
\newcommand{\epmat}{\end{pmatrix}}
% square matrix
\newcommand{\bbmat}{\begin{bmatrix}}
\newcommand{\ebmat}{\end{bmatrix}} 





\title{MAT 150A HW01}
\author{[add your name here]}
\date{Due Tuesday, 1/16/24 at 11:59 pm on Gradescope}

\begin{document}

\maketitle


\paragraph{Instructions}
Solve the following problems, and then type up your solutions in full sentences after the 
\begin{verbatim}
    \solution
\end{verbatim}
command following each exercise. It may help to look at how I typed the exercise, e.g.\ to learn the command used to typeset a particular symbol. Compile often. 
See the instructions in HW00 if you're unsure how to use Overleaf.


\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.

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}

\paragraph{Grading} Most (parts of) problems will be graded for completion out of 5 points.
A few selected problems will be graded more carefully; these will be revealed after the homework is due.
Abridged solutions will be posted after the 24-hour grace period after the homework due date. 


\paragraph{Greek letters} In mathematics (and many sciences), we require many more symbols than the English alphabet. Greek letters are used very often, so please familiarize yourself with the Greek alphabet if you haven't seen symbols like $\rho$ or $\tau$ before:
\begin{center}
\url{https://en.wikipedia.org/wiki/Greek_alphabet}
\end{center}
In LaTeX, the command for a Greek letter (capital or lowercase) is always a backslash followed by the English spelling of that letter. For example, capital \textit{Gamma} and lower case \textit{gamma} are written $\Gamma$ and $\gamma$, respectively. 




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
% you can do algebra with groups
Let $G$ be a group. Since every element in the group has an inverse, we can ``do algebra'' in the group, in the sense that we can solve equations:

\begin{proposition}
    (Cancellation Law)
    Let $a,b,c \in G$. If $ab=ac$ or $ba =ca$, then $b=c$. If $ab =a$ or if $ba = a$, then $b = 1$.
\end{proposition}

Let $x,y,z,w \in G$. 

\bea
\item Solve for $y$, given that $xyz\inv w= 1$.
\item Suppose that $xyz = 1$. Does it follow that $yzx = 1$? Does it follow that $yxz = 1$?
\ee


\solution

\bea

\item %add your solution to part (a) here

\item %add your solution to part (b) here

\ee





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

Let's recall how remainders show up when dividing integers.
Since 
\[
    32 = 6\cdot 5 + 2,
\]
we say that \emph{32 divided by 5} is \emph{6 remainder 2}. In modular arithmetic, we only care about the remainder, 2. 
Another way this is commonly written is
\[
    32 \equiv 2 \mod 5 
    \qquad
    \text{or}
    \qquad
    32 \equiv 2 \pmod{5}.
\]




\bea

\item Recall that, as a set, $\Z/5\Z = \{ 0,1,2,3,4\}$. 
 Fill out the rest of the addition table provided below. This is called the \emph{group table} for the group $(\Z/5\Z, +)$. 
 
\item Let $\Z/5\Z^\times$ denote the set $\{1,2,3,4\} = \Z/5\Z-\{0\}$. It turns out that we can define a multiplication group operation $\bullet$ using remainders as well. That is, for $x,y \in \Z/5\Z^\times$, $x \bullet y$ is just $xy \pmod{5}$ (where $x$ and $y$ are first multiplied as integers). For example, $3 \cdot 4 = 12$, and $12$ modulo $5$ is $2$, so $3 \bullet 4 = 2$ in $\Z/5\Z^\times$.

Fill out the group table for the group $(\Z/5\Z^\times, \bullet)$ provided below.

\ee

\solution

\renewcommand{\arraystretch}{1.3}

\bea

\item Group table for $(\Z/5\Z, +)$:

% replace the question marks in the table below

\begin{tabular}{|c||c|c|c|c|c|}
\hline
    +   & 0 & 1 & 2 & 3 & 4     \\
\hline
\hline
    0   & 0 & 1 & 2 & 3 & 4     \\
\hline
    1   & 1 & 2 & 3 & 4 & 0     \\
\hline
    2   & ? & ? & ? & ? & ?     \\
\hline
    3   & ? & ? & ? & ? & ?     \\
\hline
    4   & ? & ? & ? & ? & ?     \\
\hline
\end{tabular}


\item Group table for $(\Z/5\Z^\times, \bullet)$:

% replace the question marks in the table below

\begin{tabular}{|c||c|c|c|c|}
\hline
    $\bullet$   & 1 & 2 & 3 & 4     \\
\hline
\hline
    1   & 1 & 2 & 3 & 4     \\
\hline
    2   & 2 & 4 & 1 & 3     \\
\hline
    3   & ? & ? & ? & ?     \\
\hline
    4   & ? & ? & ? & ?     \\
\hline
\end{tabular}

\ee


%%%%%%%%%%%%%%%%%%%%%%
\exercise
Prove that if $x^2 = 1$ for all $x \in G$, then $G$ is abelian.

\solution

% add your solution here


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
% harder; cyclic groups?

For a real number $r \in \R$, the \textbf{floor} of $r$, denoted $\floor{r}$, is the greatest integer less than or equal to $r$.
For example, $\floor{0.3} = 0$, $\floor{65.\bar 3} = 65$, $\floor{\pi} = 3$, and $\floor{-0.8} = -1$. \\



Let $G = \{ x \in \R \st 0 \leq x < 1\}$, and for $x \in G$, let $x \star y$ be the fractional part of the real number $x+y$, i.e.\ 
\[
    x \star y = x + y - \floor{x + y}.
\]
\bea
\item Prove that $(G, \star)$ is a group, and that it is abelian.
\item Prove that $(G,\star)$ is \textbf{not} a cyclic group. (Hint: Use cardinality.)
\ee

\begin{remark}
    This group $G$ is known as the \textit{real numbers modulo 1}, in analogy with the construction of $\Z/n\Z$, the integers modulo $n$.
\end{remark}

\solution

\bea

\item %add your solution to part (a) here

\item %add your solution to part (b) here

\ee


%%%%%%%%%%%%%%%%%%%%%%
\exercise
Let $G$ be a group, and let $g_1, g_2, \ldots, g_n$ be a $n$ elements of $G$. Prove that 
\[
    (g_1 g_2 \cdots g_n)\inv = g_n\inv g_{n-1}\inv \cdots g_2\inv g_1\inv.
\]
Hint: First prove this holds for $n=2$, and then use induction.

\solution

% add your solution here








\end{document}
