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\pagestyle{fancy} \lhead{{\sf Homework 8 $|$ Due October 21 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 8510 $|$ Abstract Algebra $|$
    Fall 2022 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-6mm}

\begin{enumerate}

\item Consider homomorphisms $g_i\colon G\to H$ and $h_i\colon H\to K$ between groups, for $i=1,2$.
  \begin{enumerate}
    \item Show that if $g$ is surjective, then it right-cancels: \quad
    $
    h_1\circ g=h_2\circ g\quad\Longrightarrow\quad h_1=h_2.
    $
    \[ 
    \begin{tikzcd}[column sep=6mm, row sep=7mm]
      G \arrow[rr,two heads,"f"]\arrow[dr,two heads,"g"'] && K \\
      & H \ar[ur,dashed,two heads,"h_1"']
    \end{tikzcd}
    \hspace{8mm}\text{and}\hspace{8mm}
    \begin{tikzcd}[column sep=6mm, row sep=7mm]
      G \arrow[rr,two heads,"f"]\arrow[dr,two heads,"g"'] && K \\
      & H \ar[ur,dashed,two heads,"h_2"']
    \end{tikzcd}   
    \hspace{8mm}\Longrightarrow\hspace{10mm} h_1=h_2.
    \]
  \item Show that if $h$ is injective, then it left-cancels: \quad
    $
    h\circ g_1=h\circ g_2\quad\Longrightarrow\quad g_1=g_2.
    $
    \[ 
    \begin{tikzcd}[column sep=6mm, row sep=7mm]
      G \arrow[rr,hook,"f"]\arrow[dr,dashed,hook,"g_1"'] && K \\
      & H \ar[ur,hook,"h"']
    \end{tikzcd}
    %%
    \hspace{8mm}\text{and}\hspace{8mm}
    %% 
    \begin{tikzcd}[column sep=6mm, row sep=7mm]
      G \arrow[rr,hook,"f"]\arrow[dr,dashed,hook,"g_2"'] && K \\
      & H \ar[ur,hook,"h"']
    \end{tikzcd}
    \hspace{8mm}\Longrightarrow\hspace{10mm} g_1=g_2.
    \]
  \item Give explicit examples to show how both of the previous results can fail
    if the hypotheses are not met. \medskip
  \end{enumerate}
  

\item We defined the ascending central series via iterative ``maximal
  central descents.'' Given $N\normaleq G$, the \emph{maximal central
    descent} $[G,N]$ is characterized as being
  \[
  \text{``\emph{the smallest subgroup $L$ such that $N/L$ is central in $G/L$.}''}
  \]
 \[
  \begin{tikzpicture}[node distance=1cm,shorten >= 0, shorten <= 0,scale=1.7]
    \begin{scope}[shift={(0,0)}]
    \draw [lightestred,fill=lightestred] (0,2.18) to[bend left=70] (0,-.15) to[bend left=60] (0,2.18);
    \node (G) at (0,3.5) {$G$};
    \node (Z) at (0,2.75) {$\Balert{Z}$};
    \node (N) at (0,2) {$\bm{N}$};
    \node (H) at (0,1) {$H$};
    \node (L) at (0,0) {$\hspace{37mm}\Alert{\bm{L=[G,N]=[G,Z]}}$};
    %\node (GH) at (0,-.625) {$[G,H]$};
    %\node (GL) at (0,-1.25) {$[G,L]$};
    %\node (1) at (0,-1.875) {$\<1\>$};
    %%
    %\draw [-stealth',very thick] (-1.3,2) to[bend left] (-.3,2.1);
    \draw [-stealth',very thick,Red]  (N) to[bend right=45] (L);
    \draw [-stealth',very thick,dotted,Red]  (Z) to[bend right=20] (L);
    %\draw [-stealth',very thick,dotted,Red]  (H) to[bend right=40] (GH);
    %\draw [-stealth',very thick,dotted,Red]  (L) to[bend right=40] (GL);
      %%
    \node at (-1.4,1.17) {\small \Alert{central}};
    \node at (-1.4,.93) {\small \Alert{descents}};
      \draw[decorate,decoration={brace,amplitude=8pt}] 
      (-.7,.1) --  (-.7,2.1); 
      %%
    \draw [-stealth',very thick] (-1.4,0) to (-.6,0);
    \node at (-2.3,.12) {\Alert{\emph{maximal}}};
    \node at (-2.3,-.12) {\Alert{\emph{central descent}}};
      %%
    \draw (G) to (-1,3); \draw (G) to (1,3);
    \draw (G) to (Z); \draw (Z) to (N); 
    \draw (N) to (H); 
    \draw (H) to (L); 
    %\draw (L) to (GH);
    %\draw (GH) to (GL); \draw (GL) to (1);
    %\draw (1) to (-.5,-.8); \draw (1) to (.5,-.8);
    \end{scope}
    %%
    \begin{scope}[shift={(4.75,0)}]
    \draw [lightestblue,fill=lightestblue] (.05,2.9) to[bend left=60] (.05,-.15) to[bend left=70] (.05,2.9);
    \draw [lightred,fill=lightestred] (0,2.18) to[bend left=70] (0,-.15) to[bend left=60] (0,2.18);
    \node (G) at (0,3.5) {$G/L$};
    \node (Z) at (0,2.75) {$\hspace{18mm}\Balert{Z/L=Z(G/L)}$};
    \node (N) at (0,2) {$\bm{N/L}$};
    \node (H) at (0,1) {$H/L$};
    \node (L) at (0,0) {$\Alert{\bm{L/L}}$};
    %%
    \draw (G) to (-1,3); \draw (G) to (1,3);
    \draw (G) to (Z); \draw (Z) to (N); 
    \draw (N) to (H); \draw (H) to (L); 
    %\draw (1) to (-.5,-.8); \draw (1) to (.5,-.8);
    \node at (-1.7,1.495) {\small \Balert{central}};
    \node at (-1.7,1.255) {\small \Balert{subgroups}};
      \draw[decorate,decoration={brace,amplitude=8pt}] 
      (-.9,-.1) --  (-.9,2.85); 
    \end{scope}
\end{tikzpicture}
\]
Prove that $(L,\lambda)$ satisfies the following co-universal
property, where $L=[G,N]$ and $\lambda\colon G\onto G/L$ is the
canonical quotient. 
\begin{quote}
  ``\emph{If $N\normaleq G$ and $f\colon G\to K$ for which $f(N)$ is central,
    then $f$ uniquely factors through the canonical quotient map
    $\lambda\colon G\to G/L$, where $L=[G,N]$. That is, there is a unique
    homomorphism $h\colon G/L\to K$ for which $f=h\circ \lambda$.}''
\end{quote}
  \[
  \begin{tikzpicture}[scale=1.7]
    \begin{scope}[shift={(0,0)}]
      \node (G) at (-1,1) {$G$};
      \node (GN) at (0,0) {$G/L$};
      \node (K) at (1,1) {$K$};
      \draw[->>] (G) -- (GN) node[midway,below left]{\footnotesize $\lambda$};
      \draw[->] (G) -- (K) node[midway,above]{\footnotesize $f$};
      \draw[->,dashed] (GN) -- (K) node[midway,below right]{\footnotesize $h$};
    \end{scope}
    \begin{scope}[shift={(3.5,0)}]
      \node (G) at (-1,1) {$x$};
      \node (U) at (0,0) {$xL$};
      \node (U') at (1,1) {$f(x)$};
      \draw[|->] (G) -- (U) node[midway,below left]{\footnotesize $\lambda$};
      \draw[|->] (G) -- (U') node[midway,above]{\footnotesize $f$};
      \draw[|->,dashed] (U) -- (U') node[midway,below right]{\footnotesize $h$};
    \end{scope}
  \end{tikzpicture}
  \]

  
\item A \emph{universal pair} $(U,\upsilon)$ for a group $G$ with respect to a particular property consists of a group $U$ with an outgoing map $\upsilon\colon U\to G$, such that every other map $f\colon H\to G$ with that same property factors through $\upsilon$ uniquely. That is, there is a unique homomorphism $g\colon H\to U$ between the \emph{domains} such that $f=\upsilon\circ g$.
\[
\begin{tikzpicture}[scale=1.7]
  \begin{scope}[shift={(0,0)}]
    \node (G) at (1,1) {$G$};
    \node (U) at (0,0) {$U$};
    \node (H) at (-1,1) {$H$};
    \draw[->] (U) -- (G) node[midway,below right]{\footnotesize $\upsilon$};
    \draw[->] (H) -- (G) node[midway,above]{\footnotesize $f$};
    \draw[->,dashed] (H) -- (U) node[midway,below left]{\footnotesize $g$};
  \end{scope}
\end{tikzpicture}
\]
Prove that if $G$ has a universal pair $(U,\upsilon)$ with 
with respect to some property, then $U$ is unique up to isomorphism. \\



\item The \emph{product} of $\big\{G_\alpha\mid \alpha\in I\big\}$ is
  a group $P$ with a family of homomorphisms $\big\{\pi_\alpha\colon
  P\to G_\alpha\mid \alpha\in I\big\}$, satisfying the following
  universal property:
  \begin{quote}
    ``\emph{Given any group $H$ and homomorphisms $f_\alpha\colon H\to G_\alpha$, there is a unique homomorphism $g\colon H\to P$ such that $\pi_\alpha\circ g=f_\alpha$ for all $\alpha\in I$.}''
  \end{quote}
  \[
  \begin{tikzpicture}[scale=1.7,baseline=(current bounding  box.center)]]
      \begin{scope}[shift={(0,0)}]
        \node (H) at (-1,1) {$H$};
        \node (product) at (0,0) {$P$};
        \node (factors) at (1,1) {$G_\alpha$};
        \draw[->,dashed] (H) -- (product) node[midway,below left]{\footnotesize $g$};
        \draw[->] (H) -- (factors) node[midway,above]{\footnotesize $f_\alpha$};
        \draw[->>] (product) -- (factors) node[midway,below right]{\footnotesize $\pi_\alpha$};
      \end{scope}
  \end{tikzpicture}
  \]
  %%
  \begin{enumerate}
  \item Show that $Z(\prod_\alpha G_\alpha)=\prod_\alpha Z(G_\alpha)$.
  \item Show that $(G_1\times\cdots\times G_n)'=G'_1\times\cdots\times G'_n$.
  \end{enumerate}
  

  
\item The \emph{co-product} of $\big\{G_\alpha\mid \alpha\in I\big\}$
  is a group $S$ with a family of homomorphisms
  $\big\{\iota_\alpha\colon G_\alpha\to S\mid \alpha\in I\big\}$,
  satisfying the following co-universal property:
  \begin{quote}
    ``\emph{Given any group $H$ and homomorphisms $f_\alpha\colon G_\alpha\to H$, there is a unique homomorphism $h\colon S\to H$ such that $h\circ\iota_\alpha=f_\alpha$ for all $\alpha\in I$.}''
  \end{quote}
  \[
  \begin{tikzpicture}[scale=1.7,baseline=(current bounding  box.center)]]
      \begin{scope}[shift={(0,0)}]
        \node (factors) at (-1,1) {$G_\alpha$};
        \node (sum) at (0,0) {$S$};
        \node (H) at (1,1) {$H$};
        \draw[->] (factors) -- (sum) node[midway,below left]{\footnotesize $\iota_\alpha$};
        \draw[->] (factors) -- (H) node[midway,above]{\footnotesize $f_\alpha$};
        \draw[->,dashed] (sum) -- (H) node[midway,below right]{\footnotesize $h$};
      \end{scope}
  \end{tikzpicture}
  \]
  \begin{enumerate}
    \item Show that if a non-empty family of groups $\{G_\alpha\mid
      \alpha\in I\}$ has a co-product, then it is unique up to
      isomorphism, and each $\iota_\alpha$ is injective.
    \item Prove that the set of finite sums $\sum_{\alpha\in I} A_\alpha$
      is a coproduct in the category of abelian groups, where
      $\iota_\alpha$ are the canonical injections.
  \end{enumerate}
      


  
\end{enumerate}

\end{document}
