\documentclass[12pt]{article}
\usepackage{fullpage}
\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{flushright}
  Your Name Here
\end{flushright}




\begin{center}
  \textbf{Math 71}\\ Homework 5
\end{center}



\begin{itemize}

\item[p101. \#7] Let $M$ and $N$ be normal subgroups of a group $G$
  such that $G = MN$.  Show that
  $G/(M\cap N) \cong (G/M)\times (G/N)$.

\item [p101: \# 9] Let $p$ be a prime and $G$ a group of order $p^am$
  where $p\nmid m$.  Assume that $P$ is a subgroup of $G$ having order
  $p^a$, and that $N$ is a normal subgroup of $G$ of order $p^bn$ for
  $p \nmid n$.  Prove that $|P\cap N| = p^b$ and $|PN/N| = p^{a-b}$.

  
\item [p122: \#8] Let $G$ be a group and $H$ a subgroup with
  $[G:H] = n$.  Show that $H$ contains a subgroup $K$ (normal in $G$)
  which satisfies $[G:K] \le n!$.


\item [p122: \#14] Let $G$ be a finite group with composite order $n$
  with the property that $G$ has a subgroup of order $k$ for each
  positive integer $k$ dividing $n$. Show that $G$ is not simple.

  
\item [p130: \#5] Prove that if the center of $G$ has index $n$, then
  every conjugacy class in $G$ has at most $n$ elements.

  
\item [p130: \#6] Assume that $G$ is a nonabelian group of order 15.
  Show that $|Z(G)| = 1$.  Then use the fact that
  $\la g\ra \le C_G(g)$ for all $g$ to show there is at most one class
  equation for $G$.

\item [p130: \#8] Prove that $Z(S_n) = \{1\}$ for all $n \ge 3$.

  

\end{itemize}

 

\end{document}


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