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%\definecolor{midgrey}{rgb}{.35, .35, .35} %\definecolor{amethyst}{rgb}{0.6 0.4 0.8} \definecolor{amethyst}{rgb}{0.54 0.17 0.89} \definecolor{amber}{rgb}{1.0 0.49 0.0} \definecolor{turq}{rgb}{0.12 0.69 0.71} \colorlet{lightestgray}{gray!15!white} \colorlet{lightestblue}{blue!15!white} \colorlet{midblue}{blue!45!white} \colorlet{lightestred}{red!15!white} \colorlet{midred}{red!45!white} \colorlet{lightestgreen}{darkgreen!15!white} \colorlet{lightestpurple}{amethyst!20!white} % COLORS FOR MULTIPLICATION TABLES \colorlet{tred}{red!70!white} \colorlet{tblue}{blue!60!white} %(higher number is more faded) \colorlet{tgreen}{green!75!black} %(higher number is brighter) \colorlet{tyellow}{yellow!60!white} \colorlet{tlime}{green!80!white} \colorlet{tpink}{pink} \colorlet{tpurple}{lpurple} \colorlet{torange}{orange} \newcommand*{\n}{5} % counter for Cayley tables %% CAYLEY DIAGRAM GLOBAL DEFAULTS \tikzstyle{v} = [circle, draw, fill=lightgrey,inner sep=0pt, minimum size=6mm] \tikzstyle{r} = [draw, very thick, Red, -stealth] % Red ---> \tikzstyle{b} = [draw, very thick, Blue, -stealth] % Blue ---> \tikzstyle{g} = [draw, very thick, darkgreen, -stealth] % Green ---> \tikzstyle{p} = [draw, very thick, amethyst, -stealth] % Purple ---> \tikzstyle{rr} = [draw, very thick, Red] % Red ---- \tikzstyle{bb} = [draw, very thick, Blue] % Blue ---- \tikzstyle{gg} = [draw, very thick, darkgreen] % Green ---- \tikzstyle{oo} = [draw, very thick, orange] % Purple ---- \tikzstyle{pp} = [draw, very thick, amethyst] % Purple ---- \tikzstyle{tt} = [draw, very thick, turq] % Turquoise ---- \tikzstyle{yy} = [draw, very thick, Gold] % Gold ---- \tikzstyle{R} = [draw, very thick, Red, -stealth'] % Red ---> \tikzstyle{B} = [draw, very thick, Blue, -stealth'] % Blue ---> \tikzstyle{G} = [draw, very thick, darkgreen, -stealth'] % Green ---> \tikzset{r/.style={draw, very thick, Red, -stealth}} % Red ---> \tikzset{R/.style={draw, very thick, Red, -stealth'}} % Red ---> \tikzset{v/.style={circle, draw, fill=lightgrey,inner sep=0pt, minimum size=6mm}} \renewcommand\labelitemi{\rule[0.12em]{0.4em}{0.4em}} \renewcommand\labelitemii{\normalfont\bfseries \rule[0.15em]{0.3em}{0.3em}} \renewcommand\labelitemiii{\textasteriskcentered} \renewcommand\labelitemiv{\textperiodcentered} \renewcommand{\footrulewidth}{1pt} \newenvironment{proof}[1][Proof.]{\smallskip\begin{trivlist} \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip} \newenvironment{sol}[1][Solution.]{\smallskip\begin{trivlist} \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip} \newenvironment{fminipage}% {\setlength{\fboxsep}{15pt}\begin{Sbox}\begin{minipage}}% {\end{minipage}\end{Sbox}\fbox{\TheSbox}} \newcommand{\qed}{\hfill \ensuremath{\Box}} \newcommand{\disp}{\displaystyle} \pagestyle{fancy} \lhead{{\sf Homework 11 $|$ Due April 28 (Friday) }} \rhead{\thepage} \cfoot{{\sf Math 4130 $|$ Visual Algebra $|$ Spring 2023 $|$ M.~Macauley}} \begin{document} $\;$ \vspace{-6mm} \begin{enumerate} \item Every group $G=\$ satisfying the relations $a^4=1,ドル $b^2=1,ドル and $(ab)^2=1$ is isomorphic to a quotient of $D_4$. Formally, this means that there is a unique homomorphism $D_4\onto G$ making the following diagram commute: \[ \begin{tikzpicture}[scale=1.8,baseline=(current bounding box.center)]] \begin{scope}[shift={(0,0)}] \node (S) at (-1,1) {$S$}; \node (F) at (0,0) {$D_4$}; \node (G) at (1,1) {$G$}; \draw[right hook->] (S) -- (F) node[midway,below left]{\footnotesize $\iota$}; \draw[right hook->] (S) -- (G) node[midway,above]{\footnotesize $\theta$}; \draw[->>,dashed] (F) -- (G) node[midway,below right]{\footnotesize $\pi$}; \end{scope} \end{tikzpicture} \] where $\iota$ and $\theta$ are the inclusion maps $a\mapsto a$ and $b\mapsto b$. 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. \[ \hspace*{-10mm} \newcommand\Hd{9.75} % height of C_8 \newcommand\Hc{6.5} % height of C_4 \newcommand\Hb{3.25} % height of C_2 \newcommand\Ha{0} % height of C_1 \begin{tikzpicture}[scale=1,yscale=1.3] \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,yscale=.77] \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,yscale=.77] \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,yscale=.77] %\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,yscale=.77] %\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} \] Create an analogous figure for $C_4\rtimes C_4=\$. \newpage %%---------------------------------------------------------------- \item Consider the ``mystery group'' $M=\$ defined by the following presentation. \[ M=\big\. \] The relators of this presentations describe the following motifs that a Cayley graph for $M=\$ 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'' $F=\$ of order 16ドル$ 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} %%---------------------------------------------------------------- \item Determine which group is described by each presentation, and prove that your answer is correct. %\begin{multicols}{2} \begin{enumerate} \item $G=\$ \item $G=\$ \item $G=\$ \item $G=\$. \end{enumerate} %\end{multicols} \end{enumerate} \end{document}

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