\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
\newcommand{\Frac}{\mathrm{Frac}} % fraction field
\newcommand{\normal}{\unlhd} % normal subgroup of

\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 HW03}
\author{[ADD YOUR NAME HERE]}
\date{Due Tuesday, 10/24/23 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}
Parts of Chp.\ 2, esp.\ \S 2.5--2.12.
Homomorphisms, isomorphisms, cosets, index of subgroups, the Correspondence Theorem, product groups, quotient groups.


\paragraph{Grading} Since Exam 1 is on Wednesday, October 25, all parts-of-problems in this homework will be graded out of 2 points. The goal is to practice and understand as many problems as you can. I advise against spending a disproportionate amount of time on any one problem; ask for help before the exam!

%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
Let $\varphi: G \to H$ be an \emph{isomorphism}. Prove that for all $g \in G$, the order of $g$ is the same as the order of $\varphi(g)$: $\ord(g) = \ord(\varphi(g))$. 


\solution



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
Let $(A,\star)$ and $(B, \diamond)$ be groups, and let $A\times B$ be their direct product. Recall that multiplication (i.e.\ the group operation, law of composition) is defined by 
\[
    (a_1,b_1)(a_2,b_2) = (a_1 \star a_2, b_1 \diamond b_2)
\]
for $a_i \in A$, $b_i \in B$, $i = 1,2$. 
In this exercise, you will verify all the group axioms for $A \times B$.
\bea
    \item Prove that multiplication is associative.
    \item What's the identity element $A \times B$? (Prove it.)
    \item What's the inverse of $(a,b) \in A \times B$? (Prove it.)
\ee


\solution



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
\bea
\item Let $p$ be a prime number. How many automorphisms does the cyclic group $C_p$ have?
\item How many automorphisms does $C_{24}$ have?
\ee

\solution

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
Let $K$ and $H$ be subgroups of a group $G$.
\bea
\item Prove that the intersection $K \cap H$ is a subgroup of $G$.
\item Prove that if $K \normal G$, then $K \cap H \normal H$. 
\ee

\solution



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
Prove that in a group, the products $ab$ and $ba$ are conjugate elements.

\solution





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
Prove that every subgroup of index 2 is a normal subgroup. 


\solution





\end{document}
