
\documentclass[12pt]{article}
\usepackage{myfullpage}
\usepackage{amsfonts}
\usepackage{amsmath, amsthm, amssymb}

\usepackage{enumitem}
%\pagestyle{headings}

\usepackage{etoolbox, environ}
\newenvironment{solnlist}%
   {\begin{list}{}%
                {\setlength\rightmargin{0pt}\setlength\leftmargin{0pt}}%
    \item[\hspace{2em}
         \textbf{Solution:}\quad]}
    {\end{list}}

\NewEnviron{solution}{%
  \iftoggle{printsolutions}
     {\begin{solnlist} \BODY\end{solnlist}}
     {}%
  }


\newtoggle{printsolutions}
\toggletrue{printsolutions}
%\togglefalse{printsolutions}



\newcommand{\0}{\mathbf 0}
\newcommand{\C}{{\mathbb C}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\Z}{{\mathbb Z}}
\newcommand{\diag}{\textrm{diag}}
\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}
\newcommand{\normal}{\trianglelefteq}
\newcommand{\bmid}{\,\Big|\,}
\newcommand{\ds}{\displaystyle}
\newcommand{\ms}{\medskip}
\newcommand{\bs}{\bigskip}
\newcommand{\lcm}{\mbox{lcm}}
\newcommand{\ol}{\overline}

% \begin{solution}
 %    \begin{enumerate} [label=(\alph*)]
 %    \item x
 %    \end{enumerate}
 %  \end{solution}



\begin{document}

\thispagestyle{empty}

\begin{center}
  \textbf{Math 71}\\ Homework 3
\end{center}



\begin{itemize}

\item[p48. \#2] Show the following subsets are not subgroups:
  \begin{enumerate} [label=(\alph*)]
  \item The set of 2-cycles in $S_n$ for $n \ge 3$.
  \item The set of reflections in
    $D_{2n}$.
  \item For a composite positive integer $n>1$, the set
    $\{x \in G \mid |x| = n\} \cup \{1\}$.
  \item The set of positive and negative odd integers in $\Z$ together
    with 0.
  \item The set of real numbers whose square is in $\Q$.
  \end{enumerate}


\item[p48. \#8] Let $H$ and $K$ be subgroups of a group $G$.  Show
  that $H \cup K$ is a subgroup of $G$ iff $H \subseteq K$ or
  $K \subseteq H$.

\item[p49. \#14] Show that $H= \{x\in D_{2n}\mid x^2 = 1\}$ is not a
  subgroup of $D_{2n}$ ($n \ge 3)$.

\item [p60. \#1] Find all the subgroups of $Z_{45} = \la x \ra$,
  giving a generator for each.  Describe the containments between the
  subgroups.


\item[p61. \#12] Prove that the following groups are not cyclic:
  \begin{enumerate}[label=(\alph*)]
  \item $Z_2\times Z_2$.
  \item $Z_2\times \Z$.
  \item $\Z\times \Z$.
     
  \end{enumerate}

\item [p60. \#16] Assume $|x| = m$ and $|y| = n$ and that $x,y$
  commute.  Prove that $|xy|$ divides $\lcm(m,n)$.  Need this be true
  if $x$,$y$ do not commute?  Give an example of commuting elements
  $x$,$y$ for which the order $|xy|$ is not equal to $\lcm(m,n)$.

\item [p. 85 \#1] Let $\varphi: G \to H$ be a homomorphism of groups
  and $E \leq H$.  Show that $\varphi^{-1}(E)$ is a subgroup of $G$
  and is normal in $G$ if $E$ is normal in $H$.  Conclude that
  $\ker \varphi \normal G$.

\item [p85. \#3] Let $A$ be an abelian group, and $B \le A$.  Prove
  that $A/B$ is abelian.  Give an example of a nonabelian group $G$
  containing a proper normal subgroup $N$, such that $G/N$ is abelian.

  
\item [p85. \#5] Let $G$ be a group with normal subgroup $N$, and
  consider the quotient group $G/N$.  An easy induction using the
  group law shows that $(gN)^k = g^kN$ for all integers $k$.  We are
  to show that the order $n$ of $gN$ is the smallest positive integer
  with $g^n \in N$; this is often a useful fact.  Give an example to
  show that the order of $gN$ can be strictly smaller than $|g|$.

 
\item [p86. \#10] Let $\varphi: \Z/8\Z \to \Z/4\Z$ be given by
  $\varphi(\ol a) = \ol a$, that is $\varphi([a]_8) = [a]_4$.  Show
  that $\varphi$ is a well-defined, surjective homomorphism, and
  describe its fibers and kernel explicitly.

  
\end{itemize}

% \end{itemize}


 

\end{document}


%%% Local Variables:
%%% mode: latex
%%% TeX-master: t
%%% End:
