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\pagestyle{fancy} \lhead{{\sf Homework 8 $|$ Due October 26 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 8510 $|$ Abstract Algebra I $|$
    Fall 2018 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{0mm}

\begin{enumerate}


\item Let $S=\{a\}$.
\begin{enumerate}
\item Verify that $(\N,+)$ is a free semigroup on $S$, where
  $\N=\{1,2,3,\dots\}$. 
\item Show that the free group on $S$ is an infinite cyclic group. \\
\end{enumerate}



\item Let $(F,\phi)$ be a free group on a set $S$.
  \begin{enumerate}
  \item Show that $F=\<\phi(S)\>$.
  \item Prove that every nonidentity element has infinite order 
  \item If $|S|\geq 2$, show that $F$ is not abelian.
  \item For any $T\subseteq S$, show that there exists a normal
    subgroup $N\normal F$ such that $(F/N,\pi\phi|_T)$ is a free group on
    $T$.
  \item Show that $(\<\phi(T)\>,\phi|_T)$ is a free group on $T$. \\
  \end{enumerate}



\item Let $F$ be a free group and $N$ be the subgroup generated by
  $\{x^n\mid x\in F,\; n\mbox{ a fixed integer}\}$. Show that
  $N\normal F$. \\



\item 
  Let $F$ be a free object on $S$, and $F'$ a free object on $S'$ in a
  concrete category $\mathfrak{C}$. Prove that if $|S|=|S'|$, then $F$
  and $F'$ are equivalent. \\



\item For a positive integer $n$, let $\Nil$ be the category of
  nilpotent groups, and let $\Nil_{\leq n}$ be the category of
  nilpotent groups of class at most $n$.
  \begin{enumerate}
  \item Show that there cannot exist a nilpotent group $N$ generated
    by two elements with the property that every nilpotent group
    generated by two elements is a homomorphic image of $N$.
  \item Prove or disprove that free objects always exist in $\Nil$.
  \item Prove or disprove that free objects always exist in $\Nil_{\leq n}$.
  \end{enumerate}
  

\end{enumerate}

\end{document}

