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\pagestyle{fancy} \lhead{{\sf Homework 3 $|$ Due February 7 (Monday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Spring 2022 $|$ M.~Macauley}}

\begin{document}

%$\;$

\begin{enumerate}

  %%-------------------------------------------------------------------
  
\item Below are two Cayley diagrams for the symmetric group
  \[
  S_4=\<{\color{Red}(1234)},{\color{Blue}(12)}\>
  =\<{\color{Blue}(12)},{\color{darkgreen}(13)},{\color{orange}(14)}\>.
  \]
  At left is a truncated octahedron, called the
  \emph{pemutohedron}. At right is the \emph{Nauru graph}.   
  \tikzstyle{v-tiny} = [circle, draw, fill=lightgrey,inner sep=0pt,
    minimum size=2.5mm]
  \tikzstyle{v-yel} = [circle, draw,
    fill=tyellow,inner sep=0pt, minimum size=2.5mm]
  \[
  \begin{tikzpicture}[scale=.9]
    \begin{scope}[shift={(0,0)}]
      \node (a1) at (0:1) [v-yel] {};  %% center hexigon
      \node (a2) at (60:1) [v-tiny] {};
      \node (a3) at (120:1) [v-tiny] {};
      \node (a4) at (180:1) [v-tiny] {};
      \node (a5) at (240:1) [v-tiny] {};
      \node (a6) at (300:1) [v-tiny] {}; 
      \node (b1) at (2,-.58) [v-tiny] {};  %% outer sides of 3 squares
      \node (b2) at (.5,1.87) [v-tiny] {};
      \node (b3) at (-.5,1.87) [v-tiny] {};  %% original yellow vertex
      \node (b4) at (-2,-.58) [v-tiny] {};
      \node (b5) at (-1.5,-1.44) [v-tiny] {};
      \node (b6) at (1.5,-1.44) [v-tiny] {};
      \node (c1) at (0:3) [v-tiny] {};   %% middle hexagon
      \node (c2) at (60:3) [v-tiny] {};
      \node (c3) at (120:3) [v-tiny] {};
      \node (c4) at (180:3) [v-tiny] {};
      \node (c5) at (240:3) [v-tiny] {};
      \node (c6) at (300:3) [v-tiny] {};
      \node (d1) at (0:4) [v-tiny] {};   % outer hexagon
      \node (d2) at (60:4) [v-tiny] {};
      \node (d3) at (120:4) [v-tiny] {};
      \node (d4) at (180:4) [v-tiny] {};
      \node (d5) at (240:4) [v-tiny] {};
      \node (d6) at (300:4) [v-tiny] {};
      \draw [bb] (a1) to (a2); \draw [r] (a3) to (a2); \draw [bb] (a3) to (a4);
      \draw [r] (a5) to (a4); \draw [bb] (a5) to (a6); \draw [r] (a1) to (a6);
      \draw [r] (b1) to (a1); \draw [r] (a2) to (b2); \draw [r] (b3) to (a3);
      \draw [r] (a4) to (b4); \draw [r] (b5) to (a5); \draw [r] (a6) to (b6);
      \draw [bb] (b1) to (c1); \draw [bb] (b2) to (c2); \draw [bb] (b3) to (c3);
      \draw [bb] (b4) to (c4); \draw [bb] (b5) to (c5); \draw [bb] (b6) to (c6);
      \draw [r] (c1) to (d1); \draw [r] (d2) to (c2); \draw [r] (c3) to (d3);
      \draw [r] (d4) to (c4); \draw [r] (c5) to (d5); \draw [r] (d6) to (c6);
      \draw [r] (b2) to (b3); \draw [r] (b4) to (b5); \draw [r] (b6) to (b1);
      \draw [r] (c2) to (c1); \draw [r] (c4) to (c3); \draw [r] (c6) to (c5);
      \draw [r] (d1) to (d2); \draw [bb] (d2) to (d3); \draw [r] (d3) to (d4);
      \draw [bb] (d4) to (d5); \draw [r] (d5) to (d6); \draw [bb] (d6) to (d1);
    \end{scope}
    %%
    \begin{scope}[shift={(9,0)},scale=1.75]
      \node (4321) at (0:1) [v-tiny] {};
      \node (3142) at (30:1) [v-tiny] {};
      \node (2314) at (60:1) [v-tiny] {};
      \node (1432) at (90:1) [v-tiny] {};
      \node (4213) at (120:1) [v-tiny] {};
      \node (3421) at (150:1) [v-tiny] {};
      \node (2143) at (180:1) [v-tiny] {};
      \node (1324) at (210:1) [v-tiny] {};
      \node (4132) at (240:1) [v-tiny] {};
      \node (3214) at (270:1) [v-tiny] {};
      \node (2431) at (300:1) [v-tiny] {};
      \node (1243) at (330:1) [v-tiny] {};
      %%
      \node (2341) at (0:2) [v-yel] {};
      \node (1342) at (30:2) [v-tiny] {};
      \node (4312) at (60:2) [v-tiny] {};
      \node (3412) at (90:2) [v-tiny] {};
      \node (2413) at (120:2) [v-tiny] {};
      \node (1423) at (150:2) [v-tiny] {};
      \node (4123) at (180:2) [v-tiny] {};
      \node (3124) at (210:2) [v-tiny] {};
      \node (2134) at (240:2) [v-tiny] {};
      \node (1234) at (270:2) [v-tiny] {};
      \node (4231) at (300:2) [v-tiny] {};
      \node (3241) at (330:2) [v-tiny] {};
      %%
      \draw [bb] (1432) to (2431); \draw [bb] (4213) to (3214);
      \draw [bb] (2143) to (3142); \draw [bb] (1324) to (4321);
      \draw [bb] (2413) to (3412); \draw [bb] (1342) to (2341);
      \draw [bb] (4231) to (1234); \draw [bb] (3124) to (4123);
      \draw [bb] (2314) to (4312); \draw [bb] (1243) to (3241);
      \draw [bb] (4132) to (2134); \draw [bb] (3421) to (1423);
      %%
      \draw [oo] (1432) to (4132); \draw [oo] (2314) to (3214);
      \draw [oo] (3421) to (4321); \draw [oo] (2143) to (1243);
      \draw [oo] (3412) to (4312); \draw [oo] (2134) to (1234);
      \draw [oo] (2341) to (3241); \draw [oo] (4123) to (1423);
      \draw [oo] (4213) to (2413); \draw [oo] (2431) to (4231);
      \draw [oo] (1324) to (3124); \draw [oo] (3142) to (1342);
      %%
      \draw [gg] (4213) to (1243); \draw [gg] (3421) to (2431);
      \draw [gg] (1324) to (2314); \draw [gg] (4132) to (3142);
      \draw [gg] (4312) to (1342); \draw [gg] (2134) to (3124);
      \draw [gg] (1423) to (2413); \draw [gg] (4231) to (3241);
      \draw [gg] (1432) to (3412); \draw [gg] (3214) to (1234);
      \draw [gg] (2143) to (4123); \draw [gg] (4321) to (2341);
    \end{scope}
  \end{tikzpicture}
  \]
  Carry out the following steps, taking the yellow node to represent
  the identity. \vspace{-1mm}
  \begin{enumerate}[(a)]
  \item On both diagrams, label the nodes by 
    elements of $S_4$, written in cycle notation as a 
    product of disjoint cycles.
  \item On the Nauru graph, label the nodes with permutations of the
    word $\mathbf{1234}$, where $(i\;j)$ swaps the $i^{\rm th}$ and
    $j^{\rm th}$ \emph{coordinates}.
  \item On a separate copy of the Nauru graph, label the nodes with
    permutations of $\mathbf{1234}$, where $(i\;j)$ swaps the
    \emph{numbers} $i$ and $j$. \bigskip
  \end{enumerate}

  %%-------------------------------------------------------------------

\item Two Cayley diagrams for the symmetric group $S_4$ arranged on
  flattened Archimedean solids -- the truncated cube (left) and the
  rhombicuboctahedron (right).
  %%
  \tikzstyle{v-tiny} = [circle, draw, fill=lightgrey,inner sep=0pt, 
    minimum size=2.5mm]
  \tikzstyle{v-yel} = [circle, draw, fill=yellow,inner sep=0pt, 
    minimum size=2.5mm]
  \newcommand\aaa{1}\newcommand\bbb{2}\newcommand\ccc{3}\newcommand\ddd{4.6}
  \[
  \begin{tikzpicture}
    \begin{scope}[shift={(0,0)}]
      \node (nw1) at (-1.5,1.5) [v-yel] {};
      \node (nw2) at (-.5,1.5) [v-tiny] {}; 
      \node (nw3) at (-1.5,.5) [v-tiny] {};
      \node (ne1) at (1.5,1.5) [v-tiny] {};
      \node (ne2) at (1.5,.5) [v-tiny] {}; 
      \node (ne3) at (.5,1.5) [v-tiny] {};
      \node (se1) at (1.5,-1.5) [v-tiny] {};
      \node (se2) at (.5,-1.5) [v-tiny] {};
      \node (se3) at (1.5,-.5) [v-tiny] {}; 
      \node (sw1) at (-1.5,-1.5) [v-tiny] {};
      \node (sw2) at (-1.5,-.5) [v-tiny] {}; 
      \node (sw3) at (-.5,-1.5) [v-tiny] {};
      %
      \draw [bb] (nw2) to (ne3); \draw [bb] (ne2) to (se3);
      \draw [bb] (se2) to (sw3); \draw [bb] (sw2) to (nw3);
      \draw[r](nw2)to(nw1); \draw [r] (nw3) to (nw2); \draw [r] (nw1) to (nw3);
      \draw[r](ne2)to(ne1); \draw [r] (ne3) to (ne2); \draw [r] (ne1) to (ne3);
      \draw[r](se2)to(se1); \draw [r] (se3) to (se2); \draw [r] (se1) to (se3);
      \draw[r](sw2)to(sw1); \draw [r] (sw3) to (sw2); \draw [r] (sw1) to (sw3);
      %%
      \node (NW1) at (-2.25,2.25) [v-tiny] {};
      \node (NW2) at (-3.25,2.25) [v-tiny] {}; 
      \node (NW3) at (-2.25,3.25) [v-tiny] {};
      \node (NE1) at (2.25,2.25) [v-tiny] {};
      \node (NE2) at (2.25,3.25) [v-tiny] {}; 
      \node (NE3) at (3.25,2.25) [v-tiny] {};
      \node (SE1) at (2.25,-2.25) [v-tiny] {};
      \node (SE2) at (3.25,-2.25) [v-tiny] {};
      \node (SE3) at (2.25,-3.25) [v-tiny] {}; 
      \node (SW1) at (-2.25,-2.25) [v-tiny] {};
      \node (SW2) at (-2.25,-3.25) [v-tiny] {}; 
      \node (SW3) at (-3.25,-2.25) [v-tiny] {};
      %%
      \draw [bb] (NW3) to (NE2); \draw [bb] (NE3) to (SE2);
      \draw [bb] (SE3) to (SW2); \draw [bb] (SW3) to (NW2);
      %
      \draw[r](NW2)to(NW1); \draw [r] (NW3) to (NW2); \draw [r] (NW1) to (NW3);
      \draw[r](NE2)to(NE1); \draw [r] (NE3) to (NE2); \draw [r] (NE1) to (NE3);
      \draw[r](SE2)to(SE1); \draw [r] (SE3) to (SE2); \draw [r] (SE1) to (SE3);
      \draw[r](SW2)to(SW1); \draw [r] (SW3) to (SW2); \draw [r] (SW1) to (SW3);
      %%
       \draw [bb] (nw1) to (NW1); \draw [bb] (ne1) to (NE1);
       \draw [bb] (se1) to (SE1); \draw [bb] (sw1) to (SW1);
    \end{scope}
    %%
    \begin{scope}[shift={(9,0)}]
      \node (ne-1) at (45:\aaa) [v-tiny] {};
      \node (ne-2) at (67.5:\bbb) [v-tiny] {};
      \node (ne-3) at (22.5:\bbb) [v-tiny] {};
      \node (ne-4) at (67.5:\ccc) [v-tiny] {};
      \node (ne-5) at (22.5:\ccc) [v-tiny] {};
      \node (ne-6) at (45:\ddd) [v-tiny] {};
      %%
      \node (nw-1) at (45+90:\aaa) [v-yel] {};
      \node (nw-2) at (67.5+90:\bbb) [v-tiny] {};
      \node (nw-3) at (22.5+90:\bbb) [v-tiny] {};
      \node (nw-4) at (67.5+90:\ccc) [v-tiny] {};
      \node (nw-5) at (22.5+90:\ccc) [v-tiny] {};
      \node (nw-6) at (45+90:\ddd) [v-tiny] {};
      %%
      \node (se-1) at (45-90:\aaa) [v-tiny] {};
      \node (se-2) at (67.5-90:\bbb) [v-tiny] {};
      \node (se-3) at (22.5-90:\bbb) [v-tiny] {};
      \node (se-4) at (67.5-90:\ccc) [v-tiny] {};
      \node (se-5) at (22.5-90:\ccc) [v-tiny] {};
      \node (se-6) at (45-90:\ddd) [v-tiny] {};
      %%
      \node (sw-1) at (45+180:\aaa) [v-tiny] {};
      \node (sw-2) at (67.5+180:\bbb) [v-tiny] {};
      \node (sw-3) at (22.5+180:\bbb) [v-tiny] {};
      \node (sw-4) at (67.5+180:\ccc) [v-tiny] {};
      \node (sw-5) at (22.5+180:\ccc) [v-tiny] {};
      \node (sw-6) at (45+180:\ddd) [v-tiny] {};
      %%
      \node (se-1) at (-45:1) [v-tiny] {};
      %%
      \draw [r] (ne-2) to (ne-1); \draw [r] (ne-3) to (ne-2);
      \draw [r] (ne-1) to (ne-3);
      \draw [b] (ne-2) to (ne-4); \draw [b] (ne-5) to (ne-3);
      \draw [r] (ne-4) to (ne-5); \draw [r] (ne-6) to (ne-4);
      \draw [r] (ne-5) to (ne-6);
      %%
      \draw [r] (nw-2) to (nw-1); \draw [r] (nw-3) to (nw-2);
      \draw [r] (nw-1) to (nw-3);
      \draw [b] (nw-2) to (nw-4); \draw [b] (nw-5) to (nw-3);
      \draw [r] (nw-4) to (nw-5); \draw [r] (nw-6) to (nw-4);
      \draw [r] (nw-5) to (nw-6);
      %%
      \draw [r] (se-2) to (se-1); \draw [r] (se-3) to (se-2);
      \draw [r] (se-1) to (se-3);
      \draw [b] (se-2) to (se-4); \draw [b] (se-5) to (se-3);
      \draw [r] (se-4) to (se-5); \draw [r] (se-6) to (se-4);
      \draw [r] (se-5) to (se-6);
      %%
      \draw [r] (sw-2) to (sw-1); \draw [r] (sw-3) to (sw-2);
      \draw [r] (sw-1) to (sw-3);
      \draw [b] (sw-2) to (sw-4); \draw [b] (sw-5) to (sw-3);
      \draw [r] (sw-4) to (sw-5); \draw [r] (sw-6) to (sw-4);
      \draw [r] (sw-5) to (sw-6);
      %%
      \draw [b] (ne-1) to (nw-1); \draw [b] (se-1) to (ne-1);
      \draw [b] (sw-1) to (se-1); \draw [b] (nw-1) to (sw-1);
      %%
      \draw [b] (nw-6) to (ne-6); \draw [b] (sw-6) to (nw-6);
      \draw [b] (se-6) to (sw-6); \draw [b] (ne-6) to (se-6);
      %%
      \draw [b] (ne-4) to (nw-5); \draw [b] (se-4) to (ne-5);
      \draw [b] (sw-4) to (se-5); \draw [b] (nw-4) to (sw-5);
      %%
      \draw [b] (nw-3) to (ne-2); \draw [b] (sw-3) to (nw-2);
      \draw [b] (se-3) to (sw-2); \draw [b] (ne-3) to (se-2);
    \end{scope}
  \end{tikzpicture}
  \]
  Determine what generating sets will yield these Cayley
  diagrams. Then, label the nodes with permutations in cycle notation,
  written as a product of disjoint cycles. %Put the identity at the
  %yellow node. \bigskip

  %%-------------------------------------------------------------------
  

  
\item The \emph{alternating} group $A_4$ is the subgroup of $S_4$
  that consists of the even permutations. Two Cayley diagrams are
  shown below, for presentations
  \[
  A_4=\big\<{\color{Red}(123)},{\color{Blue}(12)(34)}\big\>
  =\big\<{\color{Red}(123)},{\color{amethyst}(234)}\big\>.
  \]
  Label the nodes of these diagrams with elements of $A_4$ in cycle
  notation, written as a product of disjoint cycles.
  %Put the identity element at the yellow node.
  
  \tikzstyle{v-tiny} = [circle, draw, fill=lightgrey,inner sep=0pt,
    minimum size=3mm]
  \tikzstyle{v-yel} = [circle, draw,
    fill=tyellow,inner sep=0pt, minimum size=3mm]
  \[
  \begin{tikzpicture}[scale=.8,auto]
    \begin{scope}[shift={(0,0)}]
      \node (l1) at (0,2.25) [v-tiny] {};
      \node (l2) at (.866,3.75) [v-tiny] {};
      \node (l3) at (-.866,3.75) [v-tiny] {};
      \node (t1) at (0,1) [v-yel] {};
      \node (t2) at (.866,-.5) [v-tiny] {};
      \node (t3) at (-.866,-.5) [v-tiny] {};
      \node (r1) at (-3.68,-1.125) [v-tiny] {};
      \node (r2) at (-2.814,-2.5) [v-tiny] {};
      \node (r3) at (-1.948,-1.125) [v-tiny] {};
      \node (m2) at (1.948,-1.125) [v-tiny] {};
      \node (m1) at (3.68,-1.125) [v-tiny] {};
      \node (m3) at (2.814,-2.5) [v-tiny] {};
      \draw [bb] (l2) to (m1);
      \draw [bb] (m2) to (t2);
      \draw [bb] (r2) to (m3);
      \draw [bb] (l1) to (t1);
      \draw [bb] (l3) to (r1);
      \draw [bb] (r3) to (t3);
      \draw [r] (l1) to (l2);
      \draw [r] (l2) to (l3);
      \draw [r] (l3) to (l1);
      \draw [r] (t1) to (t3);
      \draw [r] (t2) to (t1);
      \draw [r] (t3) to (t2);
      \draw [r] (r1) to (r2);
      \draw [r] (r2) to (r3);
      \draw [r] (r3) to (r1);
      \draw [r] (m1) to (m2);
      \draw [r] (m2) to (m3);
      \draw [r] (m3) to (m1);
      \node at (0,-.2) {};
    \end{scope}
    %%
    \begin{scope}[shift={(10,.6)},scale=1.1]
      \node (a2) at (-45:1) [v-tiny] {};
      \node (a4) at (-135:1) [v-tiny] {};
      \node (a6) at (-225:1) [v-yel] {};
      \node (a8) at (-315:1) [v-tiny] {};
      \node (b1) at (0:2) [v-tiny] {};
      \node (b3) at (-90:2) [v-tiny] {};
      \node (b5) at (-180:2) [v-tiny] {};
      \node (b7) at (-270:2) [v-tiny] {};
      \node (c2) at (-45:4) [v-tiny] {};
      \node (c4) at (-135:4) [v-tiny] {};
      \node (c6) at (-225:4) [v-tiny] {};
      \node (c8) at (-315:4) [v-tiny] {};
      \draw [r] (b1) to (a8); \draw [r] (a8) to (a2); \draw [r] (a2) to (b1);
      \draw [r] (a4) to (a6); \draw [r] (a6) to (b5); \draw [r] (b5) to (a4);
      \draw [r] (c2) to (b3); \draw [r] (b3) to (c4); \draw [r] (c4) to (c2);
      \draw [r] (c6) to (b7); \draw [r] (b7) to (c8); \draw [r] (c8) to (c6);
      \draw [p] (b1) to (c2); \draw [p] (c2) to (c8); \draw [p] (c8) to (b1);
      \draw [p] (b5) to (c6); \draw [p] (c6) to (c4); \draw [p] (c4) to (b5);
      \draw [p] (b3) to (a2); \draw [p] (a2) to (a4); \draw [p] (a4) to (b3);
      \draw [p] (a6) to (a8); \draw [p] (a8) to (b7); \draw [p] (b7) to (a6);
    \end{scope}
  \end{tikzpicture}
  \]

  \bigskip


  %%-------------------------------------------------------------------
  
\item Draw the Cayley diagram of the group $G=\<a,b,c\mid
  a^2=b^3=c^3=abc=1\>$ on the skeleton of the icosahedron, shown
  below, and label the nodes with elements written using $a$, $b$,
  and $c$.
  
  \tikzstyle{v-tiny} = [circle, draw, fill=lightgrey,inner sep=0pt, 
    minimum size=4mm]
  \tikzstyle{gr} = [draw, thick, darkgrey]
  \[
  \begin{tikzpicture}[scale=1.6,auto]
    \tikzstyle{every node}=[font=\footnotesize]
    \begin{scope}[shift={(0,0)}]
      \node (m1) at (90:2.75) [v-tiny] {$a$};
      \node (m2) at (90:1) [v-tiny] {$1$};
      \node (m3) at (-90:.5) [v-tiny] {};
      \node (m4) at (-90:1) [v-tiny] {};
      \node (l1) at (210:2.75) [v-tiny] {};
      \node (l2) at (210:1) [v-tiny] {};
      \node (l3) at (150:1) [v-tiny] {$b$};
      \node (l4) at (150:.5) [v-tiny] {};
      \node (r1) at (330:2.75) [v-tiny] {};
      \node (r2) at (330:1) [v-tiny] {};
      \node (r3) at (30:1) [v-tiny] {};
      \node (r4) at (30:.5) [v-tiny] {};
      \draw [gr] (m1) to (r1); \draw [gr] (r1) to (r3); \draw [gr] (r3) to (m1);
      \draw [gr] (m2) to (l4); \draw [gr] (l4) to (l3); \draw [gr] (l3) to (m2);
      \draw [gr] (m3) to (r4); \draw [gr] (r4) to (r2); \draw [gr] (r2) to (m3);
      \draw [gr] (m4) to (l1); \draw [gr] (l1) to (l2); \draw [gr] (l2) to (m4);
      \draw [gr] (m1) to (l3); \draw [gr] (l3) to (l1); \draw [gr] (l1) to (m1);
      \draw [gr] (m2) to (r3); \draw [gr] (r3) to (r4); \draw [gr] (r4) to (m2);
      \draw [gr] (m3) to (l2); \draw [gr] (l2) to (l4); \draw [gr] (l4) to (m3);
      \draw [gr] (m4) to (r2); \draw [gr] (r2) to (r1); \draw [gr] (r1) to (m4);
      \draw [gr] (l2) to (l3); \draw [gr] (r2) to (r3);
      \draw [gr] (m1) to (m2); \draw [gr] (m3) to (m4);
      \draw [gr] (l1) to (r1); \draw [gr] (l4) to (r4);  
    \end{scope}
  \end{tikzpicture}
  \]
  There are five groups of order $12$: the abelian groups $C_{12}$ and
  $C_6\times C_2$, the dihedral group $D_6$, the alternating group
  $A_4$, and the dicyclic group $\Dic_6$. Determine which group $G$ is
  isomorphic to, and then re-draw this Cayley diagram with the nodes
  labeled with elements of that group. \bigskip
       
  %%-------------------------------------------------------------------

    
\item Prove that if $g^2=e$ for all $g\in G$, then $G$ must be
  abelian.

  %%-------------------------------------------------------------------

  
\end{enumerate}

\end{document}

