%%% 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}}




















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

\newcommand\NAME{Max Kutler}  % your name
\newcommand\HWNUM{Template}  % the homework number
\newcommand\DUEDATE{8/26/2019}  % 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]
Let $G$ be a group and $a, b \in G$. Show that $(ab)^{-1} = b^{-1} a^{-1}$.
\end{problem}

\begin{proof}
We must show that
\[ (ab) \cdot (b^{-1} a^{-1}) = e = (b^{-1} a^{-1}) \cdot (ab). \]
By associativity and the definition of inverses, we have
\[ (ab) (b^{-1} a^{-1}) = a (b b^{-1}) a^{-1} = a e a^{-1} = a a^{-1} = e. \]
Similarly,
\[ (b^{-1} a^{-1}) (ab) = b^{-1} (a^{-1} a) b = b^{-1} e b = b^{-1} b = e, \]
as desired.
\end{proof}


% \newpage % You may wish to start a new page for each exercise


\begin{problem}[2]
Give an example of a group $G$ for which the set
\[ T = \{ g \in G \mid |g| < \infty \} \]
of \textbf{torsion elements} is not a subgroup.
\end{problem}

\begin{proof}[Solution] % for exercises which do not require a proof
Let $G = \GL_2(\RR)$ be the group of invertible $2 \times 2$ matrices with entries in $\RR$. Let
\[
A =
\begin{pmatrix}
	1 & 1 \\
	0 & -1
\end{pmatrix}
\qquad \text{and} \qquad
B =
\begin{pmatrix}
	1 & 0 \\
	0 & -1
\end{pmatrix}
\]
It is easily checked that $A^2 = I_2$ and $B^2 = I_2$, and therefore $A^{-1} = A$ and $B^{-1} = B$. This means that $A, B \in \GL_2(\RR)$ and, moreover, $|A| = |B| = 2$, so that $A, B \in T$. However,
\[ AB =
\begin{pmatrix}
	1 & -1 \\
	0 & 1
\end{pmatrix},
\]
and hence
\[ (AB)^n =
\begin{pmatrix}
	1 & (-1)^n \\
	0 & 1
\end{pmatrix}
\]
is not equal to $I_2$ for any integer $n > 0$. Thus $AB \notin T$, showing that $T$ is not closed under multiplication. Therefore $T$ is not a subgroup of $\GL_2(\RR)$.
\end{proof}



\end{document}
