%%% 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}}




















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

\newcommand\NAME{Name:}  % your name
\newcommand\HWNUM{6}  % the homework number
\newcommand\DUEDATE{9/15/2023}  % 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 $x, g \in G$.
\begin{enumerate}
\item Prove by induction that $(g x g^{-1})^n = g x^n g^{-1}$ for every $n \in \NN$.

\item Use the result of part (a) to prove that $|g x g^{-1}| = |x|$.

[\textsc{hint:} There are two cases: Either $|x| < \infty$ or $|x| = \infty$.]

\item Deduce that $|ab| = |ba|$ for all $a, b \in G$.
\end{enumerate}
\end{problem}

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


\newpage


\begin{problem}[2]
Let
\[ H = \left\{
	\begin{pmatrix}
	a & b \\
	0 & d
	\end{pmatrix}
	\biggm\vert
	\text{$a,b,d \in \RR$ and $ad \neq 0$}
	\right\}.
\]
Prove that $H$ is a subgroup of $\GL_2(\RR)$.
\end{problem}

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


\newpage


\begin{problem}[3 (Judson 3.54)]
Let $G$ be a group and $H \leq G$ a subgroup. For a fixed $g \in G$, define
\[ g H g^{-1} = \{ g h g^{-1} \mid h \in H \}. \]
This is called the \textbf{conjugate of $H$ by $g$.} Prove that $g H g^{-1}$ is also a subgroup of $G$.
\end{problem}

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


\newpage


\begin{problem}[4]
Let $G$ be a finite group with $|G| = n > 2$. Prove that $G$ cannot have a subgroup $H$ with $|H| = n-1$.
\end{problem}

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



\end{document}