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\pagestyle{fancy} \lhead{{\sf Homework 10 $|$ Due November 4 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 8510 $|$ Abstract Algebra $|$
    Fall 2022 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-6mm}

\begin{enumerate}

  
\item A Cayley graph of the \emph{projective linear group}
  $\PSL_2(\Z)=\<A,B\>$ is shown below. \vspace{-4mm}
  
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    fill=lightgrey,inner sep=0pt, minimum size=3.5mm]
  \[
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    \draw[faded] (1,0) arc(180:75:1);
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    %%
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    \node at (-1.5,-.12) {\tiny $-\frac{3}{2}$};
    \node at (-1,-.12) {\scriptsize $-1$};
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    \node at (1.5,-.12) {\tiny $\frac{3}{2}$};    
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    %%
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    %%
    %%
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    \draw[faded] (-1,0) arc(0:180:.1);
    %%
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    \draw[faded] (0,0) arc(0:180:.1);
    %%
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    \draw[faded] (1,0) arc(0:180:.333);
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    \draw[faded] (1,0) arc(0:180:.1);
    %%
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    %%
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    \node (T) at (1,1.75) [v0] {};
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    \node (T3) at (2.25,1.75) {}; 
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    %%
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    \draw [bb] (I) to (S);
    \draw [bb] (T) to (TS);
    \draw [bb] (T2) to (T2S);
    %%
    \node (T-2ST) at (-2.3,.44) [v2] {};
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    \node (T-1ST) at (-1.3,.44) [v2] {};
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    \node (ST-1) at (.3,.44) [v2] {};
    \node (TST) at (.7,.44) [v2] {};
    \node (TST-1) at (1.3,.44) [v2] {};
    \node (T2ST) at (1.7,.44) [v2] {};
    \node (T2ST-1) at (2.3,.44) [v2] {};
    %%
    \draw [g] (T-2) to (T-1S); \draw [g] (T-1S) to (T-2ST-1); \draw [g] (T-2ST-1) to (T-2);
    \draw [g] (T-1) to (S); \draw [g] (S) to (T-1ST-1); \draw [g] (T-1ST-1) to (T-1);
    \draw [g] (I) to (TS); \draw [g] (TS) to (ST-1); \draw [g] (ST-1) to (I);
    \draw [g] (T) to (T2S); \draw [g] (T2S) to (TST-1); \draw [g] (TST-1) to (T);
    %%
    \node (T-2ST2) at (-2.3,.25) [v3] {};
    \node (T-2ST-2) at (-1.7,.25) [v3] {};
    \node (T-1ST2) at (-1.3,.25) [v3] {};
    \node (T-1ST-2) at (-.7,.25) [v3] {};
    \node (ST2) at (-.3,.25) [v3] {};
    \node (ST-2) at (.3,.25) [v3] {};    
    \node (TST2) at (.7,.25) [v3] {};
    \node (TST-2) at (1.3,.25) [v3] {};
    \node (T2ST2) at (1.7,.25) [v3] {};
    \node (T2ST-2) at (2.3,.25) [v3] {};
    %%
    \node (T-2ST3) at (-2.25,.15) [v4] {};
    \node (T-2ST-3) at (-1.75,.15) [v4] {};
    \node (T-1ST3) at (-1.25,.15) [v4] {};
    \node (T-1ST-3) at (-.75,.15) [v4] {};
    \node (ST3) at (-.25,.15) [v4] {};
    \node (ST-3) at (.25,.15) [v4] {};    
    \node (TST3) at (.75,.15) [v4] {};
    \node (TST-3) at (1.25,.15) [v4] {};
    \node (T2ST3) at (1.75,.15) [v4] {};
    \node (T2ST-3) at (2.25,.15) [v4] {};
    %%
    \node (T-2ST4) at (-2.18,.10) [v5] {};
    \node (T-2ST-4) at (-1.82,.10) [v5] {};
    \node (T-1ST4) at (-1.18,.10) [v5] {};
    \node (T-1ST-4) at (-.82,.10) [v5] {};
    \node (ST4) at (-.18,.10) [v5] {};
    \node (ST-4) at (.18,.10) [v5] {};    
    \node (TST4) at (.82,.10) [v5] {};
    \node (TST-4) at (1.18,.10) [v5] {};
    \node (T2ST4) at (1.82,.10) [v5] {};
    \node (T2ST-4) at (2.18,.10) [v5] {};  
    %%
    %\draw [r] (T2ST-4) to (T2ST-3) {};
    %\draw [r] (T2ST-3) to (T2ST-2) {};
    %\draw [r] (T2ST-2) to (T2ST-1) {};
    %\draw [r] (T2ST-1) to (T2S) {};
    %\draw [r] (T2S) to (T2ST) {};
    %\draw [r] (T2ST) to (T2ST2) {};
    %\draw [r] (T2ST2) to (T2ST3) {};
    %\draw [r] (T2ST3) to (T2ST4) {};
    %%
    %\draw [r] (TST-4) to (TST-3) {};
    %\draw [r] (TST-3) to (TST-2) {};
    %\draw [r] (TST-2) to (TST-1) {};
    %\draw [r] (TST-1) to (TS) {};
    %\draw [r] (TS) to (TST) {};
    %\draw [r] (TST) to (TST2) {};
    %\draw [r] (TST2) to (TST3) {};
    %\draw [r] (TST3) to (TST4) {};    
    %%
    %\draw [r] (ST-4) to (ST-3) {};
    %\draw [r] (ST-3) to (ST-2) {};
    %\draw [r] (ST-1) to (S) {};
    %\draw [r] (S) to (ST) {};
    %\draw [r] (ST) to (ST2) {};
    %\draw [r] (ST2) to (ST3) {};
    %\draw [r] (ST3) to (ST4) {};        
    %%
    %\draw [r] (T-1ST-4) to (T-1ST-3) {};
    %\draw [r] (T-1ST-3) to (T-1ST-2) {};
    %\draw [r] (T-1ST-2) to (T-1ST-1) {};
    %\draw [r] (T-1ST-1) to (T-1S) {};
    %\draw [r] (T-1S) to (T-1ST) {};
    %\draw [r] (T-1ST) to (T-1ST2) {};
    %\draw [r] (T-1ST2) to (T-1ST3) {};
    %\draw [r] (T-1ST3) to (T-1ST4) {};            
    %%
    %\draw [r] (T-2ST-4) to (T-2ST-3) {};
    %\draw [r] (T-2ST-3) to (T-2ST-2) {};
    %\draw [r] (T-2ST-2) to (T-2ST-1) {};
    %\draw [r] (T-2ST-1) to (T-2S) {};
    %\draw [r] (T-2S) to (T-2ST) {};
    %\draw [r] (T-2ST) to (T-2ST2) {};
    %\draw [r] (T-2ST2) to (T-2ST3) {};
    %\draw [r] (T-2ST3) to (T-2ST4) {};            
    %%%
    \draw [bb] (T-2ST-1) to (T-1ST) {};
    \draw [bb] (T-1ST-1) to (ST) {};
    \draw [bb] (ST-1) to (TST) {};
    \draw [bb] (TST-1) to (T2ST) {};
    %%
    \node [v35] (T-2ST-2S) at (-1.57,.17) {};
    \node [v35] (T-1ST2S) at (-1.43,.17) {};
    %\draw [r] (T-1ST2S) to (T-2ST-2S);
    \draw [bb] (T-2ST-2S) to (T-2ST-2);
    \draw [bb] (T-1ST2) to (T-1ST2S);
    %%
    \node [v35] (T-1ST-2S) at (-.57,.17) {};
    \node [v35] (ST2S) at (-.43,.17) {};
    %\draw [r] (ST2S) to (T-1ST-2S);
    \draw [bb] (T-1ST-2S) to (T-1ST-2);
    \draw [bb] (ST2) to (ST2S);
    %%
    \node [v35] (ST-2S) at (.43,.17) {};
    \node [v35] (TST2S) at (.57,.17) {};
    %\draw [r] (TST2S) to (ST-2S);
    \draw [bb] (ST-2S) to (ST-2);
    \draw [bb] (TST2) to (TST2S);
%%
    \node [v35] (TST-2S) at (1.43,.17) {};
    \node [v35] (T2ST2S) at (1.57,.17) {};
    %\draw [r] (T2ST2S) to (TST-2S);
    \draw [bb] (TST-2S) to (TST-2);
    \draw [bb] (T2ST2) to (T2ST2S);
    %%%
    \draw [g] (T-2ST-2) to (T-1ST); \draw [g] (T-1ST) to (T-1ST2S); \draw [g] (T-1ST2S) to (T-2ST-2);
    \draw [g] (T-1ST-2) to (ST); \draw [g] (ST) to (ST2S); \draw [g] (ST2S) to (T-1ST-2);
    \draw [g] (ST-2) to (TST); \draw [g] (TST) to (TST2S); \draw [g] (TST2S) to (ST-2);
    \draw [g] (TST-2) to (T2ST); \draw [g] (T2ST) to (T2ST2S); \draw [g] (T2ST2S) to (TST-2);
    %%
    \node [v6] (-2L_R) at (-2.31,.08) {};
    \node [v6] (-2R_L) at (-1.69,.08) {}; \node [v6] (-2R_M) at (-1.635,.07) {}; \node [v6] (-2R_R) at (-1.58,.09) {};
    \node [v6] (-1R_L) at (-.69,.08) {}; \node [v6] (-1R_M) at (-.635,.07) {}; \node [v6] (-1R_R) at (-.58,.09) {};
    \node [v6] (0R_L) at (.31,.08) {}; \node [v6] (0R_M) at (.365,.07) {}; \node [v6] (0R_R) at (.42,.09) {};
    \node [v6] (1R_L) at (1.31,.08) {}; \node [v6] (1R_M) at (1.365,.07) {}; \node [v6] (1R_R) at (1.42,.09) {};
    \node [v6] (-1L_R) at (-1.31,.08) {}; \node [v6] (-1L_M) at (-1.365,.07) {}; \node [v6] (-1L_L) at (-1.42,.09) {};
    \node [v6] (0L_R) at (-.31,.08) {}; \node [v6] (0L_M) at (-.365,.07) {}; \node [v6] (0L_L) at (-.42,.09) {};
    \node [v6] (1L_R) at (.69,.08) {}; \node [v6] (1L_M) at (.635,.07) {}; \node [v6] (1L_L) at (.58,.09) {};
    \node [v6] (2L_R) at (1.69,.08) {}; \node [v6] (2L_M) at (1.635,.07) {}; \node [v6] (2L_L) at (1.58,.09) {};
    \node [v6] (2R_L) at (2.31,.08) {};
    %%
    \draw [g-thin] (T-2ST-2S) to (-2R_M); \draw [bb-thin] (-2R_R) to (-2R_M); \draw [g-thin] (-2R_M) to (T-2ST-3); \draw [bb-thin] (T-2ST-3) to (-2R_L); 
    \draw [g-thin] (T-2ST-3) to (T-2ST-2S);
    \draw [g-thin] (T-1ST-2S) to (-1R_M); \draw [bb-thin] (-1R_R) to (-1R_M); \draw [g-thin] (-1R_M) to (T-1ST-3); \draw [bb-thin] (T-1ST-3) to (-1R_L); 
    \draw [g-thin] (T-1ST-3) to (T-1ST-2S);
    \draw [g-thin] (ST-2S) to (0R_M); \draw [bb-thin] (0R_R) to (0R_M); \draw [g-thin] (0R_M) to (ST-3); \draw [bb-thin] (ST-3) to (0R_L); 
    \draw [g-thin] (ST-3) to (ST-2S);
    \draw [g-thin] (TST-2S) to (1R_M); \draw [bb-thin] (1R_R) to (1R_M); \draw [g-thin] (1R_M) to (TST-3); \draw [bb-thin] (TST-3) to (1R_L); 
    \draw [g-thin] (TST-3) to (TST-2S);    
    %%
    %%
    \draw [g-thin] (T-1ST2) to (-1L_R); \draw [bb-thin] (T-1ST3) to (-1L_R); \draw [g-thin] (-1L_R) to (-1L_L); \draw [bb-thin] (-1L_M) to (-1L_L); \draw [g-thin] (-1L_L) to (T-1ST2);
    \draw [g-thin] (ST2) to (0L_R); \draw [bb-thin] (ST3) to (0L_R); \draw [g-thin] (0L_R) to (0L_L); \draw [bb-thin] (0L_M) to (0L_L); \draw [g-thin] (0L_L) to (ST2);
    \draw [g-thin] (TST2) to (1L_R); \draw [bb-thin] (TST3) to (1L_R); \draw [g-thin] (1L_R) to (1L_L); \draw [bb-thin] (1L_M) to (1L_L); \draw [g-thin] (1L_L) to (TST2);
    \draw [g-thin] (T2ST2) to (2L_R); \draw [bb-thin] (T2ST3) to (2L_R); \draw [g-thin] (2L_R) to (2L_L); \draw [bb-thin] (2L_M) to (2L_L); \draw [g-thin] (2L_L) to (T2ST2);
    %%
    \draw [g-xthin] (T-2ST-4) to (-2R_L); \draw [g-xthin] (T-1ST-4) to (-1R_L); \draw [g-xthin] (ST-4) to (0R_L); \draw [g-xthin] (TST-4) to (1R_L);
    %\draw [bb-thin] (ST3) to (0L_R); \draw [r-thin] (0L_R) to (0L_M); \draw [bb-thin] (0L_M) to (0L_L); \draw [r-thin] (0L_L) to (ST2S);
    %\draw [bb-thin] (TST3) to (1L_R); \draw [r-thin] (1L_R) to (1L_M); \draw [bb-thin] (1L_M) to (1L_L); \draw [r-thin] (1L_L) to (TST2S);
    %\draw [bb-thin] (T2ST3) to (2L_R); \draw [r-thin] (2L_R) to (2L_M); \draw [bb-thin] (2L_M) to (2L_L); \draw [r-thin] (2L_L) to (T2ST2S);
    \draw [g-thin] (T-2ST2) to (-2L_R); \draw [g-thin] (T2ST-2) to (2R_L);
    \draw [bb-thin] (T-2ST3) to (-2L_R); \draw [bb-thin] (T2ST-3) to (2R_L);
    %%
    \draw [gg] (T-2S) to (-2.3,1.072); \draw [gg] (T2) to (2.3,1.459);
  \end{tikzpicture}
  \]
  The generators are the images $A=\pi(ST)$ and $B=\pi(S)$ under the natural quotient map $\pi\colon\SL_2(\Z)\onto\PSL_2(\Z)$, where
  \[
S=\begin{bmatrix}0 & -1 \\ 1 & 0\end{bmatrix},\qquad T=\begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix},\qquad 
ST=\begin{bmatrix}0 & -1 \\ 1 & 1\end{bmatrix}.
  \]
  Thus, we can think of them as $A=\pm S$ and $B=\pm ST$. 
  In this problem, we will show that the identity element in $\PSL_2(\Z)$ cannot be written nontrivially as
  \begin{equation*}\label{eq6:PSL(2,Z)}
    I=A^{i_1}B^{j_1}A^{i_2}B^{j_2}\cdots A^{i_{m-1}}B^{j_{m-1}}A^{i_m},\qquad i_k\in\{0,1,2\},\quad j_k\in\{0,1\}.
\end{equation*}
  This will confirm that $\PSL_2(\Z)\cong C_3*C_2$, which is suggested by the Cayley graph above. 
  \begin{enumerate}
  \item Show by slight brute force that this is impossible for $m=1$ and $m=2$.
  \item Now, suppose there is such a representation of the idenitity, for some $m\geq 3$. Assuming that $m$ is minimal, left-multiply by $A^{-i_1}$ and right-multiply by $A^{i_1}$. Show that $i_m+i_0$ is not a multiple of $3$, and conclude that the identity element can be written as a product of $BA$'s and $BA^2$'s.
  \item Recalling that $A=\pm ST$ and $B=\pm S$, let $R=ST$, and consider the following matrices in $\SL_2(\Z)$:
    \[
    SR=S^2T=-T=-\begin{bmatrix} 1 & 1 \\ 0 & 1\end{bmatrix},\qquad SR^2=-TST=-\begin{bmatrix} 1 & 0 \\ 1 & 1\end{bmatrix}.
    \] 
    Show that for any (nontrivial) product of these matrices, the absolute sum of the entries is at least $3$. 
    \item Conclude that the identity element in $\PSL_2(\Z)$ cannot be written nontrivially.
\end{enumerate}
  

  \item Consider the ``mystery group'' $M=\<S_1\mid R_1\>$ defined by the following presentation.
  \[
  M=\big\<a,b,c\mid a^4,\;c^2,\;a^2b^{-2},\;aba^{-1}b^{-1},\;aca^{-1}c,\;a^2bc^{-1}b^{-1}c^{-1}\big\>.
  \]
  The relators of this presentations describe the following motifs that a Cayley graph for $M=\<a,b,c\>$ must have.
  \[
\tikzstyle{v0} = [circle, draw, fill=lightgrey,inner sep=0pt,
  minimum size=2mm]
\tikzstyle{v1} = [circle, draw, fill=black,inner sep=0pt,
  minimum size=2mm]
\begin{tikzpicture}[scale=1]
  \begin{scope}[shift={(0,0)},scale=1]
     \node [v1] (1) at (0,0) {};
     \node [v0] (r) at (1,0) {};
     \node [v0] (r2) at (1,1) {};
     \node [v0] (r3) at (0,1) {};
     \node at (.5,-1) {$a^4=1$};
     \draw [r] (1) to (r); \draw [r] (r) to (r2); \draw [r] (r2) to (r3); \draw [r] (r3) to (1);  
     \end{scope}
     %%
    \begin{scope}[shift={(2.5,0)},scale=1]
     \node [v1] (1) at (0,.5) {};
     \node [v0] (f) at (1,.5) {};
     \draw [gg] (1) to (f);      
     \node at (.5,-1) {$c^2=1$};
     \end{scope}
     %%
    \begin{scope}[shift={(5,0)},scale=1]
     \node [v1] (1) at (0,0) {};
     \node [v0] (r) at (1,0) {};
     \node [v0] (r2) at (1,1) {};
     \node [v0] (r3) at (0,1) {};
     \node at (.5,-1) {$a^2=b^{2}$};
     \draw [r] (1) to (r); \draw [r] (r) to (r2); \draw [b] (r3) to (r2); \draw [b] (1) to (r3);  
     \end{scope}
     %%
     \begin{scope}[shift={(7.5,0)},scale=1]
     \node [v1] (1) at (0,0) {};
     \node [v0] (r) at (1,0) {};
     \node [v0] (r2) at (1,1) {};
     \node [v0] (r3) at (0,1) {};
     \node at (.5,-1) {$ab=ba$};
     \draw [r] (1) to (r); \draw [b] (r) to (r2); \draw [r] (r3) to (r2); \draw [b] (1) to (r3);  
     \end{scope}
     %%
     \begin{scope}[shift={(10,0)},scale=1]
     \node [v1] (1) at (0,0) {};
     \node [v0] (r) at (1,0) {};
     \node [v0] (r2) at (1,1) {};
     \node [v0] (r3) at (0,1) {};
     \node at (.5,-1) {$ac=ca$};
     \draw [r] (1) to (r); \draw [g] (r) to (r2); \draw [r] (r3) to (r2); \draw [g] (1) to (r3);  
     \end{scope}
     
    \begin{scope}[shift={(13.5,.5)},scale=1]
     \node [v1] (1) at (0:.75) {};
     \node [v0] (r) at (60:.75) {};
     \node [v0] (r2) at (120:.75) {};
     \node [v0] (r3) at (180:.75) {};
     \node [v0] (r4) at (240:.75) {};
     \node [v0] (r5) at (300:.75) {};
     \node at (270:1.5) {$a^2b=cbc$};
     \draw [r] (1) to (r); \draw [r] (r) to (r2); \draw [b] (r2) to (r3); 
     \draw [g] (r4) to (r3); \draw [b] (r5) to (r4); \draw [g] (1) to (r5);  
     \end{scope}
\end{tikzpicture}
\]
\begin{enumerate}
\item Establish $|M|\leq 16$ by showing that every word
  in $M$ can be written
  \[
  a^ib^jc^k,\qquad i\in\{0,1,2,3\},\quad j\in\{0,1\},\quad k\in\{0,1\},
  \]
\item Identify a ``familiar group'' $F=\<S_2\mid R_2\>$ of order $16$ whose generators satisfy these relations. That is, define a ``relabing map'' $\theta\colon S_1\onto S_2$ that extends to $\theta\colon R_1\to R_2$.
\item Describe why it follows that $M\cong F$.
\end{enumerate}


% Hungerford 1.9.6, Grove 10.1(b), Grove 12.56, Hungerford 1.9.14. 
\item Prove what group is described by each presentation.
  %\begin{multicols}{2}
    \begin{enumerate}
    \item $G=\<a,b\mid a^2=1,\;b^3=1,\;ab=ba\>$
    \item $G=\<a,b\mid a^4=1,\;a^2=b^2,\;ab=ba^3\>$
    \item $G=\<a,b\mid a^4=b^3=1,\;ab=ba^3\>$
    \item $G=\<a,b,c\mid a^2=b^2=c^2=(ab)^3=(ac)^2=(bc)^3=1\>$.
    \end{enumerate}
  %\end{multicols}


  %% 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{fiber
    product} (or \emph{pullback}) 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. 

  
\item Let $A$, $B_1$, $B_2$ be objects in a category $\cat$ and let $\alpha_i\in\Hom_{\cat}(A,B_i)$ for $i=1,2$. A \emph{fiber coproduct} (or \emph{pushout}) is an object $C$ with morphisms $\iota_i\in\Hom(B_i,C)$ satisfying the following couniversal property:
\begin{quote}
\emph{For any object $D\in\Ob(\cat)$ and morphisms $h_i\in\Hom_{\cat}(B_i,D)$ such that if $h_1\circ\alpha_1=h_2\circ\alpha_2$, there exists a unique $h\in\Hom_{\cat}(C,D)$ such that $h\circ \iota_i=h_i$.}
\end{quote}
\[
\begin{tikzpicture}[scale=2.5,baseline=(current bounding  box.center)]]
\begin{scope}[shift={(0,0)}]
\node (A) at (0,1) {$A$};
\node (A1) at (1,1) {$B_1$};
\node (A2) at (0,0) {$B_2$};
\node (B) at (1,0) {$C$};
\node (C) at (1.61,-.61) {$D$};
\draw[->] (A) -- (A1) node[midway,above]{\footnotesize $\alpha_1$};
\draw[->] (A) -- (A2) node[midway,left]{\footnotesize $\alpha_2$};
\draw[->] (A1) -- (B) node[midway,right]{\footnotesize $\iota_1$};
\draw[->] (A2) -- (B) node[midway,above]{\footnotesize $\iota_2$};
%%
\draw[->,dashed] (B) -- (C) node[midway,above right]{\footnotesize $h$};
\path[->] (A1) edge [bend left] node [above right] {\footnotesize $h_1$} (C);
\path[->] (A2) edge [bend right] node [below left] {\footnotesize $h_2$} (C);
\end{scope}
\end{tikzpicture}
\]
  


  Let $Y$ and $Z$ be sets with inclusion maps $\alpha_Y\colon Y\cap
  Z\into Y$ and $\alpha_Z\colon Y\cap Z\into Z$. Show that the pushout
  (or fiber coproduct) of $\alpha_Y$ and $\alpha_Z$ is equivalent to the
  union $Y\cup Z$, as illustrated by the following commutative diagram.
  \[
  \begin{tikzpicture}[scale=2.5,xscale=1.2,baseline=(current bounding  box.center)]]
      \begin{scope}[shift={(0,0)}]
        \node (YcapZ) at (0,1) {$Y\cap Z$};
        \node (Z) at (1,1) {$Z$};
        \node (Y) at (0,0) {$Y$};
        \node (YcupZ) at (1,0) {$Y\cup Z$};
        \draw[right hook->] (YcapZ) -- (Z) node[midway,above]{\footnotesize $\alpha_Z$};
        \draw[right hook->] (YcapZ) -- (Y) node[midway,left]{\footnotesize $\alpha_Y$};
        \draw[right hook->] (Z) -- (YcupZ) node[midway,right]{\footnotesize $\iota_Z$};
        \draw[right hook->] (Y) -- (YcupZ) node[midway,above]{\footnotesize $\iota_Y$};
      \end{scope}
  \end{tikzpicture}
  \]


\end{enumerate}

\end{document}




  
\end{enumerate}

\end{document}
