\documentclass[12pt]{article}
\usepackage{fancyhdr,fancybox,amssymb,epsfig,amsmath,latexsym}
\usepackage[all]{xy}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric}
\usepackage{enumerate}
\usepackage{multicol}
\usepackage{color}
\colorlet{darkgreen}{green!70!black}
\definecolor{darkblue}{rgb}{0, 0, .6}
\definecolor{midnightblue}{rgb}{.2, .2, .7}
\definecolor{grey}{rgb}{.7, .7, .7}
\definecolor{lightgrey}{rgb}{.85, .85, .85}
\definecolor{Red}{rgb}{.9,0,0}
\definecolor{Blue}{rgb}{0,0,.9}
\definecolor{Green}{rgb}{0,.9,0}
\colorlet{lred}{red!50!white}
\definecolor{lblue}{rgb}{0.60, 0.75, 1}
\colorlet{lgreen}{green!40!white}
\colorlet{lyellow}{yellow!30!white}
\definecolor{lpurple}{rgb}{0.7 0.45 0.9}
\colorlet{lorange}{orange!50!white}

\colorlet{darkgreen}{green!70!black}
\colorlet{lightred}{red!30!white}
\colorlet{lightblue}{blue!30!white}
\colorlet{lightgreen}{darkgreen!30!white}
\definecolor{lpurple}{rgb}{0.6 0.35 0.8}
\colorlet{darkpurple}{lpurple!70!black}

\usetikzlibrary{arrows}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{intersections}

\topmargin=-.5in
\headsep=0.2in
\oddsidemargin=0in
\textwidth=6.7in
\textheight=9.2in
\footskip=.5in

\newcommand{\C}{\mathbb{C}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\id}{id}

\def\<{\langle}
\def\>{\rangle}
\def\into{\hookrightarrow}
\def\longto{\longrightarrow}
\def\normal{\lhd}
\def\normaleq{\unlhd}
\DeclareMathOperator\cl{cl}
\DeclareMathOperator\Aut{Aut}
\DeclareMathOperator\Inn{Inn}
\DeclareMathOperator\Char{char}
\DeclareMathOperator\Gal{Gal}
\DeclareMathOperator\Stab{Stab}
\DeclareMathOperator\Fix{Fix}
\DeclareMathOperator\Perm{Perm}

\renewcommand\labelitemi{\rule[0.12em]{0.4em}{0.4em}}
\renewcommand\labelitemii{\normalfont\bfseries \rule[0.15em]{0.3em}{0.3em}}
\renewcommand\labelitemiii{\textasteriskcentered}
\renewcommand\labelitemiv{\textperiodcentered}

\renewcommand{\footrulewidth}{1pt}
\newenvironment{proof}[1][Proof.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{sol}[1][Solution.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{fminipage}%
{\setlength{\fboxsep}{15pt}\begin{Sbox}\begin{minipage}}%
{\end{minipage}\end{Sbox}\fbox{\TheSbox}}
\newcommand{\qed}{\hfill \ensuremath{\Box}}
\newcommand{\disp}{\displaystyle}

\pagestyle{fancy} \lhead{{\sf Homework 4 $|$ Due September 26 (Wednesday)
}} \rhead{\thepage} \cfoot{{\sf Math 8510 $|$ Abstract Algebra I $|$
    Fall 2018 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-3mm}


\begin{enumerate}

\item For each of the following statements, prove or give a counterexample. 
\begin{enumerate}
\item Let $f\colon G\to H$ be an epimorphism. For any two
  homomorphsims $g_1,g_2\colon H\to K$, the equality $g_1\circ
  f=g_2\circ f$ implies that $g_1=g_2$.

\item Let $f\colon G\to H$ be a monomorphism. For any two
  homomorphsims $g_1,g_2\colon H\to K$, the equality $g_1\circ
  f=g_2\circ f$ implies that $g_1=g_2$.

\item Let $g\colon H\to K$ be an epimorphism. For any two
  homomorphsims $f_1,f_2\colon G\to H$, the equality $g\circ
  f_1=g\circ f_2$ implies that $f_1=f_2$.

\item Let $g\colon H\to K$ be a monomorphism. For any two
  homomorphsims $f_1,f_2\colon G\to H$, the equality $g\circ
  f_1=g\circ f_2$ implies that $f_1=f_2$. \medskip
\end{enumerate}



\item If $(U,\varepsilon)$ is a universal pair for a group $G$ and
  $h\in\Aut(U)$, show that $(U,h\varepsilon)$ is also
  universal for $G$. Conversely, if $(U,\varepsilon_1)$ is universal for $G$,
  show that $\varepsilon_1=h\varepsilon$ for some
  $h\in\Aut(U)$.~ \medskip



\item For each of the following groups $G$:
  \[
  \text{(i)}\;\;D_3,\hspace{19mm} \text{(ii)}\;\;D_4,\hspace{19mm} \text{(iii)}\;\;Q_8,\;\hspace{19mm}\text{(iv)}\;\;A_4,\;\hspace{19mm}\text{(v)}\;\;S_4,
  \] 
  carry out the steps below. Feel free to use Google for Part~(a).
  \begin{enumerate}
  \item Draw the subgroup lattice of $G$ and circle each normal
    subgroup.
  \item Determine the commutator subgroup $G'$ and double-circle this
    on the lattice.
  \item Sketch the subgroup lattice of $G/G'$ and find its isomorphism type. 
  \item Find the derived series for $G$. \medskip
  \end{enumerate}



% Grove Exercise 5.3(2,3,4)
\item Prove the lemma from class:
  \begin{enumerate}
  \item If $G'\leq H\leq G$ show that $H\lhd G$.
  \item Show that if $K\lhd G$, then $K'\lhd G$. 
  \item Suppose $f\colon G\to H$ is an epimorphism, with $\ker
    f=K$. Show that $H$ is abelian if and only if $G'\leq K$. \medskip
  \end{enumerate}



\item Prove that $G$ is solvable in the following cases, where $p$ and
  $q$ are distinct primes:
  \begin{enumerate}
  \item $|G|=p^n$,
  \item $|G|=p^2q$. \medskip
  \end{enumerate}



\item Let $G$ be a finite group and $N\normaleq G$.
  \begin{enumerate}
  \item If the order of $xN\in G/N$ is a power of $p$, show that there
    exists $y\in G$ such that $|y|$ is a power of $p$ and $yN=xN$.
  \item If $G/N$ is abelian and $P$ is a Sylow $p$-subgroup of $G$,
    prove that $PN/N$ is the unique Sylow $p$-subgroup of $G/N$.
  \item Show by example how Part~(b) can fail if $G/N$ is non-abelian.
  \end{enumerate} 

\end{enumerate}

\end{document}

