%%% Template for Math 4580 homework, after Eric J. Malm's hmcpset class
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%%% PREAMBLE -----------------------------------------------------------
% Do make any changes before \begin{document}, except to add macros
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\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
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\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{8}  % the homework number
\newcommand\DUEDATE{9/26/2023}  % 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 $G$ be a group, and let $H \leq G$ and $K \leq G$ be subgroups.
\begin{enumerate}
\item Prove that $H \cap K \leq G$.

\item Prove that $H \cup K \leq G$ if and only if either $H \subseteq K$ or $K \subseteq H$.
\end{enumerate}
\end{problem}

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


\newpage


\begin{problem}[2]
Let $S$ be a set, and let $G$ be the set of all bijections $f \colon S \to S$. It follows from Math 3345 that $G$ is a group under function composition $\circ$. Indeed
\begin{itemize}
\item \textbf{$\circ$ is a binary operation on $G$:} If $f \colon S \to S$ and $g \colon S \to S$ are bijections, then $f \circ g \colon S \to S$ is a bijection.
\item \textbf{$\circ$ is associative:} $(f \circ g) \circ h = f \circ (g \circ h)$.
\item \textbf{Identity:} The identity function $\id_S \colon S \to S$, defined by $\id_S(x) = x$ for all $x \in S$, is a bijection and it is the identity element for $\circ$.
\item \textbf{Inverses:} For any bijection $f \colon S \to S$, the inverse function $f^{-1} \colon S \to S$ is also a bijection, and it is the inverse of $f$ with respect to $\circ$.
\end{itemize}

You don't need to prove any of these statements, but they should all sound familiar! \\

In the questions below, we consider the case where $S = \{1, 2, 3\}$.
\begin{enumerate}
\item Compute $|G|$.
\item Find $f, g \in G$ such that $f \circ g \neq g \circ f$. Thus, $G$ is a non-abelian group.
\item Show that if $f \in G$ is not the identity, then either $f^2 = \id_S$ or $f^3 = \id_S$.
\end{enumerate}
\end{problem}

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



\end{document}