\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}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\QQ}{\mathbb{Q}}




\begin{document}

\pagestyle{empty}

\begin{center}
{\large MATH 150A Winter 2020 \ -  \ {\bf Problem Set 2}}

% Put your name and section here.
{\large due January 24}
\end{center}


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

	\item
	\begin{enumerate}
	 \item Characterize the elements of $\CC^\times$ that have order $n$ for positive integer $n$.
	 \item Characterize the elements of $\CC^\times$ that have order $\infty$.
	\end{enumerate}

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

	\item (2.4.3) Let $a$ and $b$ be elements of a group $G$.  Prove that $ab$ and $ba$ have the same order.

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

	\item (2.4.10) Show by example that the product of elements of finite order in a group need not have finite order.  What if the group is abelian?

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

	\item (2.5.2) Let $H$ and $K$ be subgroups of group $G$.
	\begin{enumerate}
	 \item Prove that the intersection $K \cap H$ is a subgroup of $H$.
	 \item Prove that if $K$ is a normal subgroup of $G$, then $K \cap H$ is a normal subgroup of $H$.
	\end{enumerate}
	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

	\item 
	\begin{enumerate}
	 \item Find an injective homomorphism from the symmetric group $S_3$ to $\GL_3(\RR)$.
	 \item Let $C_8$ denote the cyclic group of order 8.  Find an injective homomorphism from $C_8$ to $\GL_2(\RR)$.
	\end{enumerate}

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

	\item (2.5.4) Let $f:\RR^+ \to \CC^\times$ be the map defined by $f(x) = e^{ix}$.  Prove that $f$ is a homomorphism, and determine its kernel and image.

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

	\item Let $D_5$ denote the {\em dihedral group of the pentagon}, which is the group of order 10 consisting of the symmetries of a regular pentagon in the plane.  $D_5$ is generated by $r$ and $s$ which represent a counter-clockwise rotation of the pentagon by $2\pi/5$ radians, and a reflection, respectively.  Find all subgroups of $D_5$ and determine which subgroups are normal.

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

	\item 
	 \begin{enumerate}
	  \item Prove that if $f:\QQ^+ \to \QQ^+$ is a group homomorphism, then $f(x) = cx$ for some constant $c$.
	  \item Let $V$ and $W$ be vector spaces over $\QQ$ and $T:V\to W$ a function.  Prove that $T$ is a group homomorphism between $(V,+)$ and $(W,+)$ if and only if $T$ is a linear map.
	  \item Is the property in part (a) true for $f:\CC^+ \to \CC^+$?
	  \item Is the property in part (a) true for $f:\RR^+ \to \RR^+$?
	  \end{enumerate}

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


