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\pagestyle{fancy} \lhead{{\sf Homework 9 $|$ Due October 31 (Tuesday)
}} \rhead{\thepage} \cfoot{{\sf Math 8510 $|$ Abstract Algebra $|$
    Fall 2023 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-6mm}

\begin{enumerate}


\item Every group $G=\<a,b\>$ satisfying the relations $a^4=1$, $b^2=1$, and $(ab)^2=1$ is isomorphic to a quotient of $D_4$. Said differently, $D_4$ is the \emph{largest} group on two generators that satisfies these relations. The figure below shows all of the distinct ways to collapse the Cayley diagram for $D_4$ by right cosets of a subgroup, and the correponding relation(s) added, if the result is a group. 
\begin{enumerate}
\item Characterize $D_4$ by a univeral or co-universal property involving relations, and include the correct commutative diagram. 
\item Create an analogous figure for $D_6=\<a,b\mid a^6=b^2=(ab)^2=1\>$.
\item Create an analogous figure for $D_6=\<s,t\mid s^2=t^2=(st)^6=1\>$.
\end{enumerate}


  \[
  \hspace*{-10mm}
\newcommand\Hd{9.75} % height of C_4 
\newcommand\Hc{6.5} % height of C_3  
\newcommand\Hb{3.25} % height of C_2 
\newcommand\Ha{0} % height of C_1   
  \begin{tikzpicture}[scale=1]
    \tikzstyle{v-r} = [circle, draw, fill=lightgrey,inner sep=0pt,
    minimum size=2mm]
\tikzstyle{v-g} = [circle, draw, fill=lightgrey,inner sep=0pt,
    minimum size=2mm]
\tikzstyle{v-b} = [circle, draw, fill=lightgrey,inner sep=0pt,
    minimum size=2mm]
\tikzstyle{v-p} = [circle, draw, fill=lightgrey,inner sep=0pt, 
    minimum size=2mm]
\tikzstyle{v-o} = [circle, draw, fill=lightgrey,inner sep=0pt, 
    minimum size=2mm]
\tikzstyle{v-gr} = [circle, draw, fill=black,inner sep=0pt,
    minimum size=2mm]
    \tikzstyle{v-tiny} = [circle, draw, fill=lightgrey,inner sep=0pt,
    minimum size=2mm]
    \tikzstyle{lat} = [draw,very thick,lightgrey]
      \tikzstyle{every node}=[font=\small]
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \begin{scope}[shift={(9,0)},scale=1]
       \tikzset{every loop/.style={min distance=10mm,looseness=15}}
    \begin{scope}[shift={(0,\Hd)},scale=.33]
    %\draw[very thick,rounded corners,lightgrey] (-2.75,-2.25) rectangle (2.75,3.6);
      \node (1) at (0:0) [v-gr] {};
      \path (1) edge [b,in=150,out=210,loop,-stealth] (1);
      \path (1) edge [r,in=-30,out=30,loop,-stealth] (1);
      \node (D4-bot) at (0,-2) {};
      \node [anchor=east] at (6,.8) {$a=1$};
      \node [anchor=east] at (6,-.8) {$b=1$};
      %\node [anchor=east] at (5,-1.75) {$(ab)^2=1$};
    \end{scope}
    \begin{scope}[shift={(-5,\Hc)},scale=.33]
       \node (1) at (0:2) [v-gr] {};
      \node (r2) at (180:2) [v-p] {};
      \draw [rr] (1) to (r2);
      \node (r2-f-top) at (0,2.75) {};
      \node (r2-f-bot) at (0,-1.75) {};
      \path (1) edge [b,in=60,out=120,loop,-stealth] (1);
      \path (r2) edge [b,in=60,out=120,loop,-stealth] (r2);
      \node [anchor=west] at (-7.25,0) {$b=1$};
    \end{scope}
    \begin{scope}[shift={(0,\Hc)},scale=.33]
      \node (1) at (0:2) [v-gr] {};
      \node (r2) at (180:2) [v-p] {};
      \draw [rr] (1) to [bend right=20] (r2);
      \draw [bb] (1) to [bend left=20] (r2);
      \node (r-top) at (0,2.75) {};
      \node (r-bot) at (0,-1.75) {};
     %\node [anchor=east] at (8,1.75) {$a^2=1$};
     %\node [anchor=east] at (8,0) {$b^2=1$};
     \node [anchor=east] at (7.25,0) {$ab=1$};
    \end{scope}
    \begin{scope}[shift={(5,\Hc)},scale=.33]
      \node (1) at (0:2) [v-gr] {};
      \node (r2) at (180:2) [v-p] {};
      \draw [bb] (1) to (r2);
      \node (r2-rf-top) at (0,2.75) {};
      \node (r2-rf-bot) at (0,-1.75) {};
      \path (1) edge [r,in=60,out=120,loop,-stealth] (1);
      \path (r2) edge [r,in=60,out=120,loop,-stealth] (r2);
       \node [anchor=east] at (7.25,0) {$a=1$};
    \end{scope}
    %%
    \begin{scope}[shift={(-5,\Hb+.2)},scale=.33]
      \node (1) at (0:2) [v-gr] {};
      \node (r) at (90:2) [v-r] {};
      \node (r2) at (180:2) [v-p] {};
      \node (r3) at (270:2) [v-r] {};
      \draw [r] (1) to [bend right=30] (r);
      \draw [r] (r) to [bend right=30] (r2);
      \draw [r] (r2) to [bend right=30] (r3);
      \draw [r] (r3) to [bend right=30] (1);
      \draw [bb] (r) to (r3);
      \path (1) edge [b,in=-30,out=30,loop,-stealth] (1);
      \path (r2) edge [b,in=150,out=210,loop,-stealth] (r2);
      \node (f-top) at (0,3) {};
      \node (f-bot) at (0,-2.75) {};    
      \node [anchor=west] at (-10.5,0) {\emph{not a group}};
    \end{scope}
    \begin{scope}[shift={(0,\Hb+.2)},scale=.33]
      \node (1) at (0:2) [v-gr] {};
      \node (r) at (90:2) [v-r] {};
      \node (r2) at (180:2) [v-p] {};
      \node (r3) at (270:2) [v-r] {};
      \draw [rr] (1) to [bend right=30] (r);
      \draw [bb] (r) to [bend right=30] (r2);
      \draw [rr] (r2) to [bend right=30] (r3);
      \draw [bb] (r3) to [bend right=30] (1);
      \node (r2-top) at (0,3) {};
      \node (r2-bot) at (0,-2.75) {};
      \node [anchor=east] at (7.25,0) {$a^2=1$};
      %\node [anchor=east] at (8,0) {$b^2=1$};
      %\node [anchor=east] at (8,-1.75) {$(ab)^2=1$};
      \end{scope}
    \begin{scope}[shift={(5,\Hb+.2)},scale=.33]
      %\draw[very thick,rounded corners,lightgrey] (-2.75,-2.25) rectangle (2.75,3.6);
      \node (1) at (0:2) [v-gr] {};
      \node (r) at (90:2) [v-r] {};
      \node (r2) at (180:2) [v-p] {};
      \node (r3) at (270:2) [v-r] {};
      \draw [r] (1) to [bend right=30] (r);
      \draw [r] (r) to [bend right=30] (r2);
      \draw [r] (r2) to [bend right=30] (r3);
      \draw [r] (r3) to [bend right=30] (1);
      \draw [bb] (1) to [bend left=15] (r);
      \draw [bb] (r2) to [bend left=15] (r3);
      %\node [midgrey] at (-2.,-1.75) {\tiny $\times 2$};
      \node (rf-top) at (0,3) {};
      \node (rf-bot) at (0,-2.75) {};
      \node [anchor=east] at (9.75,0) {\emph{not a group}};
    \end{scope}
    \begin{scope}[shift={(0,\Ha)},scale=.33]
      %\draw[very thick,rounded corners,lightgrey] (-2.75,-2.25) rectangle (2.75,3.6);
      \node (1) at (0:2.9) [v-gr] {};
      \node (r) at (90:2.9) [v-r] {};
      \node (r2) at (180:2.9) [v-p] {};
      \node (r3) at (270:2.9) [v-r] {};
      \node (f) at (0:1.4) [v-b] {};
      \node (rf) at (90:1.4) [v-g] {};
      \node (r2f) at (180:1.4) [v-b] {};
      \node (r3f) at (270:1.4) [v-g] {};
      \draw [r] (1) to [bend right=30] (r);
      \draw [r] (r) to [bend right=30] (r2);
      \draw [r] (r2) to [bend right=30] (r3);
      \draw [r] (r3) to [bend right=30] (1);
      \draw [r] (f) to [bend left=22] (r3f);
      \draw [r] (r3f) to [bend left=22] (r2f);
      \draw [r] (r2f) to [bend left=22] (rf);
      \draw [r] (rf) to [bend left=22] (f);
      \draw [bb] (1) to (f);
      \draw [bb] (r) to (rf);
      \draw [bb] (r2) to (r2f);
      \draw [bb] (r3) to (r3f);
      %\node [midgrey] at (-2.,-1.75) {\tiny $\times 2$};
      %\node at (0,0) {\footnotesize $\<1\>$};
      \node (1-top) at (0,4) {};
      \node [anchor=east] at (9,1.75) {$a^4=1$};
      \node [anchor=east] at (9,0) {$b^2=1$};
      \node [anchor=east] at (9,-1.75) {$(ab)^2=1$};
      \draw[decorate,decoration={brace,amplitude=7pt}] (9.25,2.25) -- (9.25,-2.25);
      \node [anchor=west] at (10.5,.8) {\emph{$D_4$ is the largest group}};
       \node [anchor=west] at (10.5,-.8) {\emph{satisfying these relations}};
    \end{scope}
    \draw [lat] (D4-bot) to (r2-f-top);
    \draw [lat] (D4-bot) to (r-top);
    \draw [lat] (D4-bot) to (r2-rf-top);
    \draw [lat] (r2-f-bot) to (f-top);
    \draw [lat] (r2-f-bot) to (r2-top);
    \draw [lat] (r-bot) to (r2-top);
    \draw [lat] (r2-rf-bot) to (r2-top);
    \draw [lat] (r2-rf-bot) to (rf-top);
    \draw [lat] (f-bot) to (1-top);
    \draw [lat] (r2-bot) to (1-top);
    \draw [lat] (rf-bot) to (1-top);
    \end{scope}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \end{tikzpicture}
\]

\item Let $\cat$ be a category.
  \begin{enumerate}
  \item If $f\in\Hom_{\cat}(A,B)$ is an isomorphism, show that
    there is at most one $g\in\Hom_{\cat}(B,A)$ such that
    $g\circ f=1_A$ and $f\circ g=1_B$.
  \item Prove that any two terminal objects in
    $\cat$ are equivalent.
  \item Suppose a family of objects $\big\{A_i\mid i\in I\big\}$
    has a coproduct $S$, with morphisms $\iota_j\in\Hom(A_j,S)$. Show that if $\cat$ has a zero object, then each $\iota_j$ is a monomorphism. \\
  \end{enumerate}



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



\item Let $F$ be a free object on $S\neq\emptyset$ in a concrete category
  $\cat$.
%\begin{enumerate}
%\item Prove that $\phi\colon S\to F$ is a monomorphism. 
 Prove that if $F'$ is a free object on a set $S'$ with
  $|S|=|S'|$, then $F$ and $F'$ are equivalent. \\
%\end{enumerate}


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




\end{enumerate}

\end{document}


  %% HW 10
  %% 
\item If $A_1$, $A_2$, and $B$ are objects in a category $\cat$ with
  morphisms $\alpha_i\in\Hom_\cat(A_i,B)$, then their
  \emph{pullback} (or \emph{fiber product}) is an object $P$ with
  morphisms $\pi_i\in\Hom_\cat(P,A_i)$ such that the following
  property holds:
  \begin{center}
    \begin{quote}
      ``\emph{For any object $Q$ in $\cat$ and morphisms
      $h_i\in\Hom_{\cat}(Q,A_i)$ such that if $\pi_1\circ h_1=\pi_2\circ h_2$,
      there exists a unique morphism $h\in\Hom_{\cat}(Q,P)$
      such that $h_i=\pi_i\circ h$ for $i=1,2$.}''
    \end{quote}
  \end{center}
  \[
  \begin{tikzpicture}[scale=2.5,baseline=(current bounding  box.center)]]
      \begin{scope}[shift={(0,0)}]
        \node (B) at (0,1) {$P$};
        \node (A1) at (1,1) {$A_1$};
        \node (A2) at (0,0) {$A_2$};
        \node (A) at (1,0) {$B$};
        \node (C) at (-.61,1.61) {$Q$};
        \draw[->] (B) -- (A1) node[midway,above]{\footnotesize $\pi_1$};
        \draw[->] (B) -- (A2) node[midway,left]{\footnotesize $\pi_2$};
        \draw[->] (A1) -- (A) node[midway,right]{\footnotesize $\alpha_1$};
        \draw[->] (A2) -- (A) node[midway,above]{\footnotesize $\alpha_2$};
        %%
        \draw[->,dashed] (C) -- (B) node[midway,above right]{\footnotesize $h$};
        \path[->] (C) edge [bend left] node [above right] {\footnotesize $h_1$} (A1);
        \path[->] (C) edge [bend right] node [below left] {\footnotesize $h_2$} (A2);
      \end{scope}
  \end{tikzpicture}
  \]
  Prove that any two pullbacks are equivalent. 


  
\end{enumerate}

\end{document}
