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\pagestyle{fancy} \lhead{{\sf Homework 5 $|$ Due October 3 (Wednesday)
}} \rhead{\thepage} \cfoot{{\sf Math 8510 $|$ Abstract Algebra I $|$
    Fall 2018 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{0mm}


\begin{enumerate}


\item Prove that if a coproduct $(S,\{\iota_\alpha\})$ exists for a
  family $\{G_\alpha\mid\alpha\in A\}$ of groups, it is unique up to
  isomorphism and each $\iota_\alpha\colon G_\alpha\to S$ is injective. \\



\item Consider a category where each object $X_i$ contains an identity
  element $e_i$ (e.g., groups, abelian groups, vector spaces,
  etc.). Consider the Cartesian product $X_1\times X_2$ of two objects
  together with the maps \vspace{-1mm}
\begin{align*}
&\iota_1\colon X_1\to X_1\times X_2,\qquad \iota_1(x)=(x,e_2) \\
&\iota_2\colon X_2\to X_1\times X_2,\qquad \iota_2(x)=(e_1,x). 
\end{align*}
\begin{enumerate}
\item Prove or disprove: this is a coproduct for $\{X_1,X_2\}$
  in the category of abelian groups.
\item Prove or disprove: this is a coproduct for $\{X_1,X_2\}$
  in the category of groups. \\
\end{enumerate}



\item Let $\{A_\alpha\mid \alpha\in I\}$ be a collection of abelian
  groups, written additively.
  \begin{enumerate}
\item Prove that the set of finite sums
  \[
  \sum_{\alpha\in I}
  A_\alpha:=\big\{x_{\alpha_1}+\cdots+x_{\alpha_k}\mid x_{\alpha_i}\in
  A_{\alpha_i}\big\}
  \]
  is a coproduct in the category of abelian groups, where
  $\iota_\alpha$ are the canonical inclusion maps.
\item Show by example how the direct product $\prod_{\alpha\in I}A_\alpha$
  fails to be a coproduct in the category of abelian groups. \\
  \end{enumerate}



\item Given groups $G$ and $H$, let $S$ be the disjoint union of their
  nonidentity elements. The \emph{free product}, denoted $G*H$, is the
  set of all words of the form
  \[
  G*H=\{s_1s_2\cdots s_k\mid s_i\in S\},
  \]
  where no two consecutive terms both lie in $G$ or in $H$. You may
  assume that $G*H$ is a group where the operation is concatenation,
  and reduction by iterative removal of identity elements
  $gg^{-1}$ and $hh^{-1}$ that arise.
  \begin{enumerate}
  \item Prove that $G*H$ is a coproduct of $\{G,H\}$ in the category
    of groups.
  \item Describe the group $\Z_2*\Z_2$. What do the elements look
    like? Write out a group presentation (no proof needed) and
    construct an isomorphism to a familiar group. \\
  \end{enumerate}
  


\item 
  \begin{enumerate}
  \item Show that if $G=G_1G_2$ is an internal direct product, then $G/G_i\cong G_j$ for $i\neq j$.
  \item Show that $Z(\prod_\alpha G_\alpha)=\prod_\alpha Z(G_\alpha)$.
  \item Show that $(G_1\times\cdots\times G_n)'=G'_1\times\cdots\times G'_n$.
  \item Under what circumstances is $G_1\times\cdots\times
    G_n$ solvable? 
\end{enumerate}


\end{enumerate}


\end{document}

