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\pagestyle{fancy} \lhead{{\sf Homework 9 $|$ Due November 2 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 8510 $|$ Abstract Algebra I $|$
    Fall 2018 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{0mm}


\begin{enumerate}


\item Let $F$ be a free object on a set $S$, with $\phi\colon S\to
  F$, in a concrete category $\mathfrak{C}$. Construct a new category
  $\mathfrak{D}$, by carefully defining $\Ob(\mathfrak{D})$ and
  $\Hom(\mathfrak{D})$, so that $\phi\colon S\to F$ arises as either
  an initial (universal) object or as a terminal (couniversal)
  object. Justify your claims. \\



\item Prove what group is described by each presentation. 
\begin{enumerate}
\item $G=\<a,b\mid a^2=b^2=(ab)^2=1\>$. 
\item $G=\<a,b\mid a^4=1,\;a^2=b^2,\;ab=ba^3\>$. 
\item $G=\<a,b\mid a^2=1,\;b^3=1,\;ab=ba\>$.
\item $G=\<a,b\mid a^4=b^3=1,\;ab=ba^3\>$.
\item $G=\<a,b,c\mid a^2=b^2=c^2=(ab)^3=(ac)^2=(bc)^3=1\>$. \\
%\item $G=\<a,b\mid a^2=1,\;b^3=1\>$. \\
\end{enumerate}



\item Consider the Cartesian product $H=\Z_2\times\Z_n$ (as a
  \emph{set}) with binary operation defined as
  \[
  (\bar{i},\bar{j})\cdot(\bar{k},\bar{\ell})
  =(\bar{i}+\bar{k},(-1)^k\bar{j}+\bar{\ell}).
  \]
  \begin{enumerate}
    \item Show that $H$ is a group under this operation, and determine
      its order.
    \item Let $G=\<a,b\mid a^n=1,\;b^2=1,\; abab^{-1}=1\>$. Show
      that $|G|\leq 2n$.
    \item Show that $H\cong G$. \\
  \end{enumerate}



\item Define the \emph{generalized quaternion group} $Q_n$ by the presentation
  \[
  Q_n=\<a,b\mid a^{2n}=1,\;b^2=a^n,\;ab=ba^{-1}\>,\qquad
  \mbox{for $n\geq 1$}\;.
  \]
  \begin{enumerate}
  \item Show that $|Q_n|=4n$.
  \item Show that $Q_1$ is cyclic and $Q_2$ is the quaternion group. 
  \item Show that $Q_3$ is not isomorphic with either $D_6$ or $A_4$. 
  \item Find the nilpotency class of $Q_{2^n}$. \\
  \end{enumerate}



\item Let $S$ be a set. The group with presentation $\<S\mid R\>$
  where $R=\{[s,t]\mid s,t\in S\}$ is called the \emph{free abelian}
  group on $S$; denote it by $A_S$ and let $\phi\colon S\to A_S$ be
  the inclusion.
  \begin{enumerate}
  \item Prove that $A_S$ has the following universal property: for
    any abelian group $G$ and function $\theta\colon S\to G$, there
    is a unique homomorphism $f\colon A_S\to G$ such that
    $f\phi=\theta$.
  \item Deduce that if $A$ is a free abelian group on a set of
    cardinality $n$, then
    \[
    A\cong \Z\times\cdots\times\Z\qquad\mbox{($n$ factors)}.
    \] \vspace{-2mm} 
  \end{enumerate}



\end{enumerate}


\end{document}

