%%% 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{\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{9}  % the homework number
\newcommand\DUEDATE{10/7/2019} % 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
\[ D_4 = \langle r, s \mid r^4 = s^2 = 1, rs = sr^{-1} \rangle. \]
\begin{enumerate}
\item Find all cyclic subgroups of $D_4$.
\item Find all proper subgroups of $D_4$ which are not cyclic.
\item Draw the subgroup lattice of $D_4$.
\end{enumerate}
\end{problem}

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

\newpage


\begin{problem}[2]
Recall that
\[ Q_8 = \langle -1, i, j, k \mid (-1)^2 = 1, \ i^2 = j^2 = k^2 = ijk = -1 \rangle. \]
As a set, we have
\[ Q_8 = \{ 1, i, j, k, -1, -i, -j, -k \}, \]
where $-i$ is shorthand for $(-1) \cdot i$, and similarly for $-j$ and $-k$.
\begin{enumerate}
\item Write the multiplication table for $Q_8$.
\item Find all cyclic subgroups of $Q_8$.
\item Prove that every proper subgroup of $Q_8$ is cyclic.
\item Draw the subgroup lattice of $Q_8$.
\end{enumerate}
\end{problem}

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

\newpage


\begin{problem}[3]
Let $G$ be a group and let $g \in G$ be an element. Define a function
\begin{align*}
	\lambda_g \colon G &\to G \\
	x &\mapsto gx.
\end{align*}

\begin{enumerate}
\item Prove that $\lambda_g$ is a bijection from $G$ to itself.
\item Prove that $\lambda_g$ is an isomorphism if and only if $g$ is the identity element in $G$.
\end{enumerate}
\end{problem}

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

\newpage


\begin{problem}[4]
Let $G$ be a group. Define a relation on $G$ by
\[ x \sim y \ \text{if and only if there exists $g \in G$ such that $y = gxg^{-1}$}. \]
Prove that $\sim$ is an equivalence relation.
\end{problem}

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



\end{document}
