\documentclass[12pt]{article}

\usepackage{amssymb,amsmath,amsthm} 
\usepackage{color}
\usepackage{verbatim}

%%% Margins %%%
\setlength{\topmargin}{0pt}
\setlength{\headheight}{0pt}
\setlength{\headsep}{0pt}
\setlength{\topskip}{0pt}
\setlength{\footskip}{0pt}
\setlength{\oddsidemargin}{-20pt}
\setlength{\textheight}{9in}
\setlength{\textwidth}{6.7in}
\setlength{\parindent}{0pt}
\setlength{\parskip}{4ex}

%%% Macros %%%
% Some macros to get you started
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\SL}{SL}
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\Aut}{Aut}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\QQ}{\mathbb{Q}}
\renewcommand{\Re}{\operatorname{Re}}

\begin{document}

\pagestyle{empty}

\begin{center}
{\large MATH 150A Winter 2020 \ -  \ {\bf Problem Set 7}}

% Put your name and section here.
{\large due February 28}
\end{center}


\begin{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

	\item Find a presentation in terms of generators and relations for the following groups.
	\begin{enumerate}
	 \item $\ZZ \times \ZZ$
	 \item $C_3 \times C_3$
	 \item $S_3$
	 \item $A_4$
	\end{enumerate}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

	\item The group $G = \langle x,y \mid xyx^{-1}y^{-1} \rangle$ is called a {\em free abelian group}.  Prove a mapping property of this group: If $u$ and $v$ are elements of an abelian group $A$, there is a unique homomorphism $\varphi:G \to A$ such that $\varphi(x) = u$ and $\varphi(y) = v$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

	\item Let $F$ be the free group on $\{x,y\}$.  Prove that the elements $u = x^2$, $v = y^2$, and $z = xy$ generate a subgroup isomorphic to the free group on $\{u,v,z\}$.
	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

	\item Let $F$ be the free group on a nonempty set $S$ with $|S| = k$.  How many elements with reduced word of length $n$ does $F$ have?

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

	\item 
	\begin{enumerate}
	 \item Prove that the additive group of $\QQ$ is not finitely generated.
	 \item Prove that the multiplicative group $\QQ^\times$ is not finitely generated.
	\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{enumerate}
\end{document}


