%%% Template for Math 4580 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{\NN}{\mathbb{N}}
\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{12}  % the homework number
\newcommand\DUEDATE{10/10/2023}  % the due date
\newcommand\COURSE{Math 4580}  % 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]
Let $Q_8 = \{1, -1, i, -i, j, -j, k, -k\}$, which is a group under multiplication where
\begin{itemize}
\item $1$ is the identity element.
\item $(-1) \cdot i = -i$, and similarly for $-j$ and $-k$.
\item $i^2 = j^2 = k^2 = -1$.
\item $ij = k$, $jk = i$, and $ki = j$.
\item $ji = -k$, $kj = -i$, and $ik = -j$.
\end{itemize}
\begin{enumerate}
\item Use the above information to write the Cayley table (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}[2]
Let $G$ be a group and $x \in G$. Define the \textbf{conjugacy class} containing $x$ to be the set
\[ \{ g x g^{-1} \mid g \in G \}. \]
[\textsc{note:} On Exam 1, you proved that the relation
\[ x \sim y \ \text{if and only if there exists $g \in G$ such that $y = gxg^{-1}$} \]
is an equivalence relation on $G$. The conjugacy class containing $x$ is just the equivalence class $[x]$ under this relation.] \\

Compute all of the conjugacy classes in the following groups.
\begin{enumerate}
\item $\ZZ_8$
\item $D_4$
\item $Q_8$
\end{enumerate}
\end{problem}

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



\end{document}