%%% Template for MA 361 homework, after Eric J. Malm's hmcpset class
%
%%% PREAMBLE -----------------------------------------------------------
% Do make any changes before \begin{document}, except to add macros
%
\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{boxedminipage}
\usepackage{graphicx}
\usepackage[margin=1in]{geometry}
\usepackage{mathpazo}
\usepackage{fancyhdr}

\setlength{\headheight}{15pt}
\setlength{\parindent}{0pt}

% include this to have lists enumerated by letters
\renewcommand{\labelenumi}{(\alph{enumi})}
\renewcommand{\labelenumii}{\roman{enumii}.}
\renewcommand{\labelenumiii}{\Alph{enumiii}.}



\newcommand{\collaborators}[1]{\begin{center}\large\sffamily{I collaborated with #1}\end{center}}

\newenvironment{problem}[1][]{%
  \bigskip% put space before problem statement box %
  \noindent\begin{boxedminipage}{\columnwidth}%
  \def\@tempa{#1}%
  \ifx\@tempa\empty\else%
    \textbf{#1}\hspace{0.5em}%
  \fi%
}{%
  \end{boxedminipage}%
}



\pagestyle{fancyplain}
\lhead{\textbf{\NAME}}
\chead{\textbf{\COURSE: HW\ \HWNUM}}
\rhead{\textbf{\DUEDATE}}
\cfoot{}


%%% MACROS -----------------------------------------------------------
% You may add more macros to this list
%
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}

\newcommand{\GL}{\operatorname{GL}}
\newcommand{\SL}{\operatorname{SL}}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\id}{\operatorname{id}}




















\begin{document}
%%% NAME AND ASSIGNMENT NUMBER -----------------------------------------------------------
% Edit the values below to identify yourself and the assignment
%

\newcommand\NAME{Name:}  % your name
\newcommand\HWNUM{13}  % the homework number
\newcommand\DUEDATE{10/24/2019} % the due date
\newcommand\COURSE{MA 361}  % the course number

%%% COLLABORATORS -----------------------------------------------------------------------------------
% If you worked with other students on this assignment, acknowledge them here
% If you did not collaborate with anyone, comment out this line
%

%\collaborators{Ferdinand Georg Frobenius and Emmy Noether}


%%% PROBLEMS AND SOLUTIONS -------------------------------------------------------------------
% Include all problems statements
% For solutions, use the \begin{proof} environment
% Note that \begin{proof}[Solution] will replace the label "Proof" with "Solution", etc.
%

\begin{problem}[1]
Recall that $\GL_n(\RR)$ is the group of all $n \times n$ invertible matrices with entries in $\RR$. Recall also that the \textbf{determinant} is a function $\det \colon \GL_n(\RR) \to \RR^{\times}$ with the properties that
\begin{itemize}
\item $\det(AB) = \det(A) \det(B)$ for all $A, B \in \GL_n(\RR)$;
\item $\det(I_n) = 1$, where $I_n \in \GL_n(\RR)$ is the $n \times n$ identity matrix.
\end{itemize}
Prove that the \textbf{special linear group}
\[ \SL_n(\RR) = \{ A \in \GL_n(\RR) \mid \det(A) = 1 \} \]
is a normal subgroup of $\GL_n(\RR)$.
\end{problem}

%\begin{proof}
%
%\end{proof}

\newpage

\begin{problem}[2]
Let $H = \{1, (1 \ 2)(3 \ 4), (1 \ 3)(2 \ 4), (1 \ 4)(2 \ 3)\} \leq S_4$. (You should convince yourself that $H$ is in fact a subgroup of $S_4$; however, you do not need to prove it here.)
\begin{enumerate}
\item Prove that $H \trianglelefteq S_4$.
\item What is the cardinality of $S_4/H$? Write a complete list (with no repetitions) of the cosets of $H$ in $S_4$.
\item Write the group operation table for the quotient group $S_4/H$.
\item Give an example of a homomorphism with domain $S_4$ and kernel $H$.
\end{enumerate}
\end{problem}

%\begin{proof}
%
%\end{proof}

\newpage


\begin{problem}[3]
Let $G$ be a group and let $N$ be a normal subgroup of $G$. Prove that the cosets $xN$ and $yN$ commute in $G/N$ if and only if $x^{-1}y^{-1}xy \in N$. (The element $x^{-1}y^{-1}xy$ is called the \textbf{commutator} of $x$ and $y$ and is denoted by $[x,y]$.)
\end{problem}

%\begin{proof}
%
%\end{proof}

\newpage


\begin{problem}[4]
Let $G$ be a group. Prove that
\[ C = \langle x^{-1}y^{-1}xy \mid x, y \in G \rangle \]
is a normal subgroup of $G$, and that $G/C$ is abelian. ($C$ is called the \textbf{commutator subgroup} of $G$.)
\end{problem}

%\begin{proof}
%
%\end{proof}



\end{document}
