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\pagestyle{fancy} \lhead{{\sf Homework 9 $|$ Due October 22 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Fall 2021 $|$ M.~Macauley}}

\begin{document}

%$\;$

\begin{enumerate}

  %%--------------------------------------------------------------------
  
\item Let $A,B\leq G$ and say that $A$ normalizes $B$ (that is, $A\leq
  N_G(B)$). 
  \begin{enumerate}[(a)]
  \item Show that $AB$ is a subgroup of $G$.
  \item Show that $B\normaleq AB$ and $A\cap B\normaleq A$. 
  \item Show that $A/(A\cap B)\cong AB/B$. [\emph{Hint:} Construct a
    homomorphism $\phi\colon A\to AB/B$ that has $A\cap B$ as kernel,
    then apply the FHT.]
  \end{enumerate}

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

\item Recall that the \emph{commutator subgroup} is defined as
  \[
  G'=\big\<xyx^{-1}y^{-1}\mid x,y\in G\big\>.
  \]
  \begin{enumerate}
  \item Show that $G'$ is the intersection of all normal subgroups of $G$
    that contain the set $C:=\{aba^{-1}b^{-1}\mid a,b\in G\}$:
    \[
    G'=\bigcap_{C\subseteq N\normaleq\,G} N.
    \]
  \item Show that $G/G'$ is abelian.
  \item For each of the groups $\SD_8$, $\AGL_1(\Z_5)$, $\Dic_{10}$,
    and $\SL_2(\Z_3)$, whose subgroup lattices are shown on the
    supplemental material, carry out the following steps. Because all
    of this can be done by inspection, you need to briefly justify
    your answers.
    \begin{enumerate}[(i)]
    \item Partition the subgroups into conjugacy classes, by drawing
      dashed circles on the lattices.
    \item The \emph{derived series} of a group is defined as
      $G^{(0)}:=G$, $G^{(1)}:=G'$, and inductively, $G^{(k)}$ is the
      commutator of $G^{(k-1)}$. Mark these groups on the lattice
      until the trivial group is reached, and determine the quotient
      $G^{(i+1)}/G^(i)$ of each successive pair.
    \end{enumerate}
  \end{enumerate}

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

\item The Cayley diagram shown below describes a group $G$ that is a
  semidirect product of $V_4$ with $C_2$. In this problem, we will
  break down the construction of this into steps.
  \tikzstyle{rr-faded} = [draw, very thick, lightred]
  \tikzstyle{bb-faded} = [draw, very thick, lightblue]
  \tikzstyle{gg-faded} = [draw, very thick, lightgreen]
  \tikzstyle{v-tiny} = [circle, draw, fill=lightgrey,inner sep=0pt,
    minimum size=3.5mm]
  \[
  \begin{tikzpicture}[scale=2]
    \node [v-tiny] (100) at (0,0) {};
    \node [v-tiny] (101) at (0,1) {};
    \node [v-tiny] (111) at (1,1) {\scriptsize $1$};
    \node [v-tiny] (110) at (1,0) {};
    \node [v-tiny] (000) at (.5,.5) {};
    \node [v-tiny] (001) at (.5,1.5) {};
    \node [v-tiny] (011) at (1.5,1.5) {};
    \node [v-tiny] (010) at (1.5,.5) {};
    \draw [bb-faded] (000) to (001);
    \draw [rr-faded] (000) to (010);
    \draw [gg-faded] (100) to (000);
    \draw [rr] (100) to (101); \draw [rr] (110) to (111);
    \draw [bb] (010) to (011);
    \draw [bb] (100) to (110); \draw [bb] (101) to (111);
    \draw [rr] (001) to (011);
    \draw [gg] (111) to (011); \draw [gg] (101) to (001); 
    \draw [gg] (110) to (010);
  \end{tikzpicture}
  \]
  \begin{enumerate}
    \item Label each node on this Cayley diagram with the order of the
      corresponding element. Just from this information alone, explain
      what familiar group this is, and why.
    \item On a fresh copy of this Cayley diagram, label the nodes with
      elements of this group.
    \item Recall that $\Aut(V_4)=\<\alpha,\beta\>\cong S_3\cong D_3$, where
      \tikzstyle{n} = [inner
      sep=1pt] \tikzstyle{o} = [draw,bend left=45,orange,-stealth]
      \tikzstyle{p} = [draw,bend left=45,amethyst,-stealth]
      \[
      \begin{tikzpicture}[scale=.75]
        \tikzstyle{every node}=[font=\normalsize,anchor=south]
        \begin{scope}[shift={(0,0)}]
          \node at (0,-.12) {\normalsize ${\bm \alpha}$ :};
          \node [n] (1) at (1,0) {$h$};
          \node [n] (2) at (2,0) {$v$};
          \node [n] (3) at (3,0) {$r$};
          \path [o] (1) to (2);
          \path [o] (2) to (3);
          \path [o] (3) to (1);
        \end{scope}
        %%
        \begin{scope}[shift={(8,0)}]
          \node at (0,-.27) {\normalsize ${\bm \beta}$ :};
          \node [n] (1) at (1,0) {$h$};
          \node [n] (2) at (2,0) {$v$};
          \node [n] (3) at (3,0) {$r$};
          \path [p] (1) to (2);
          \path [p] (2) to (1);
        \end{scope}
      \end{tikzpicture}
      \]
      Let $C_2=\<s\>=\{1,s\}$. Construct the group $G$ as a semidirect
      product
      \[
      G\cong V_4\rtimes_\theta C_2=\big\{(a,b)\mid a\in V_4,\,b\in C_2\big\}.
      \]
      That is, define an explicit labeling map $\theta\colon
      C_2\to\Aut(V_4)$. [\emph{Note}: There's more than one map that
        will work.]
    \item On a fresh copy of the Cayley diagram, label each node as an
      ordered pair in $V_4\rtimes_\theta C_2$.
      
  \end{enumerate}

  %%--------------------------------------------------------------------
  
\item Recall that the automorphism group of $D_3$ is
  $\Aut(D_3)=\<\alpha,\beta\>\cong D_3$, where
  \[
  \left\{\!\!\begin{array}{ll}
    \alpha(r)=r \\ \alpha(f)=rf
  \end{array}\right.\qquad\qquad
  \left\{\!\!\begin{array}{ll}
    \beta(r)=r^2 \\ \beta(f)=f
  \end{array}\right.
  \]
  All of these automorphisms are inner, because
  $\Inn(D_3)\normaleq\Aut(D_3)$ and $\Inn(D_3)\cong D/Z(D_3)\cong
  D_3$. Two Cayley diagrams for $\Aut(D_3)$ are shown below.
  
  \tikzstyle{v-tiny} = [circle, draw, fill=grey,inner sep=0pt, minimum size=1.2mm] 
  \tikzstyle{gg-thin} = [draw, darkgreen]
  \tikzstyle{r-thin} = [draw, Red, -stealth]
  \tikzstyle{bb-thin} = [draw, Blue]
  \tikzstyle{o} = [draw, very thick,orange, -stealth]
  \tikzstyle{p} = [draw, very thick,amethyst]
  %%
  \[
  \hspace*{-10mm}
  \begin{tikzpicture}[scale=.9]
    %%%%
    %%%%%=============================================================
    %%%%
    \begin{scope}[shift={(0,0)},scale=1]
      \begin{scope}[shift={(0:1.25)},scale=.12]
        \draw[grey, fill=lightgrey] (0,0) circle (60mm);
        \node (f) at (0:2.5) [v-tiny] {};
        \node (rf) at (120:2.5) [v-tiny] {};
        \node (rrf) at (240:2.5) [v-tiny] {};
        \node (e) at (0:5.5) [v-tiny] {};
        \node (rr) at (240:5.5) [v-tiny] {};
        \node (r) at (120:5.5) [v-tiny] {};
        \draw [r-thin] (e) to [bend left] (rr);
        \draw [r-thin] (rr) to [bend left] (r);
        \draw [r-thin] (r) to [bend left] (e);
        \draw [r-thin] (f) to [bend right] (rf);
        \draw [r-thin] (rf) to [bend right] (rrf);
        \draw [r-thin] (rrf) to [bend right] (f);
        \draw [bb-thin] (e) to (f);
        \draw [bb-thin] (r) to (rf);
        \draw [bb-thin] (rr) to (rrf);
        \node (o-in_f) at (90:4.8) {};
        \node (o-out_f) at (270:4.8) {};
        \node (p_f) at (0:4.8) {};
      \end{scope}
      %%
      \begin{scope}[shift={(120:1.25)},scale=.12]
        \draw[grey, fill=lightgrey] (0,0) circle (60mm);
        \node (f) at (0:2.5) [v-tiny] {};
        \node (rf) at (120:2.5) [v-tiny] {};
        \node (rrf) at (240:2.5) [v-tiny] {};
        \node (e) at (0:5.5) [v-tiny] {};
        \node (rr) at (240:5.5) [v-tiny] {};
        \node (r) at (120:5.5) [v-tiny] {};
        \draw [r-thin] (e) to [bend left] (rr);
        \draw [r-thin] (rr) to [bend left] (r);
        \draw [r-thin] (r) to [bend left] (e);
        \draw [r-thin] (f) to [bend right] (rf);
        \draw [r-thin] (rf) to [bend right] (rrf);
        \draw [r-thin] (rrf) to [bend right] (f);
        \draw [bb-thin] (e) to [bend left] (rrf);
        \draw [bb-thin] (rr) to [bend left] (rf);
        \draw [bb-thin] (r) to [bend left] (f);
        \node (o-in_rf) at (195:4.8) {};
        \node (o-out_rf) at (30:4.8) {};
        \node (p_rf) at (120:4.8) {};
      \end{scope}
      %%
      \begin{scope}[shift={(240:1.25)},scale=.12]
        \draw[grey, fill=lightgrey] (0,0) circle (60mm);
        \node (f) at (0:2.5) [v-tiny] {};
        \node (rf) at (120:2.5) [v-tiny] {};
        \node (rrf) at (240:2.5) [v-tiny] {};
        \node (e) at (0:5.5) [v-tiny] {};
        \node (rr) at (240:5.5) [v-tiny] {};
        \node (r) at (120:5.5) [v-tiny] {};
        \draw [r-thin] (e) to [bend left] (rr);
        \draw [r-thin] (rr) to [bend left] (r);
        \draw [r-thin] (r) to [bend left] (e);
        \draw [r-thin] (f) to [bend right] (rf);
        \draw [r-thin] (rf) to [bend right] (rrf);
        \draw [r-thin] (rrf) to [bend right] (f);
        \draw [bb-thin] (e) to [bend right] (rf);
        \draw [bb-thin] (r) to [bend right] (rrf);
        \draw [bb-thin] (rr) to [bend right] (f);
        \node (o-in_r2f) at (330:4.8) {};
        \node (o-out_r2f) at (165:4.8) {};
        \node (p_r2f) at (240:4.8) {};
      \end{scope}
      %%
      %%----------------------------------------------------------------
      %%
      \begin{scope}[shift={(0:3.5)},scale=.12]
        \draw[grey, fill=lightgrey] (0,0) circle (60mm);
        \node (f) at (0:2.5) [v-tiny] {};
        \node (rf) at (120:2.5) [v-tiny] {};
        \node (rrf) at (240:2.5) [v-tiny] {};
        \node (e) at (0:5.5) [v-tiny] {};
        \node (rr) at (240:5.5) [v-tiny] {};
        \node (r) at (120:5.5) [v-tiny] {};
        \draw [r-thin] (e) to [bend right] (r);
        \draw [r-thin] (r) to [bend right] (rr);
        \draw [r-thin] (rr) to [bend right] (e);
        \draw [r-thin] (f) to [bend left] (rrf);
        \draw [r-thin] (rrf) to [bend left] (rf);
        \draw [r-thin] (rf) to [bend left] (f);
        \draw [bb-thin] (e) to (f);
        \draw [bb-thin] (r) to (rf);
        \draw [bb-thin] (rr) to (rrf);
        \node (o-in_1) at (270:4.8) {};
        \node (o-out_1) at (90:4.8) {};
        \node (p_1) at (180:4.8) {};
      \end{scope}
      %%
      \begin{scope}[shift={(120:3.5)},scale=.12]
        \draw[grey, fill=lightgrey] (0,0) circle (60mm);
        \node (f) at (0:2.5) [v-tiny] {};
        \node (rf) at (120:2.5) [v-tiny] {};
        \node (rrf) at (240:2.5) [v-tiny] {};
        \node (e) at (0:5.5) [v-tiny] {};
        \node (rr) at (240:5.5) [v-tiny] {};
        \node (r) at (120:5.5) [v-tiny] {};
        \draw [r-thin] (e) to [bend right] (r);
        \draw [r-thin] (r) to [bend right] (rr);
        \draw [r-thin] (rr) to [bend right] (e);
        \draw [r-thin] (f) to [bend left] (rrf);
        \draw [r-thin] (rrf) to [bend left] (rf);
        \draw [r-thin] (rf) to [bend left] (f);
        \draw [bb-thin] (e) to [bend right] (rf);
        \draw [bb-thin] (r) to [bend right] (rrf);
        \draw [bb-thin] (rr) to [bend right] (f);
        \node (o-in_r) at (30:4.8) {};
        \node (o-out_r) at (195:4.8) {};
        \node (p_r) at (300:4.8) {};
      \end{scope}
      %%
      \begin{scope}[shift={(240:3.5)},scale=.12]
        \draw[grey, fill=lightgrey] (0,0) circle (60mm);
        \node (f) at (0:2.5) [v-tiny] {};
        \node (rf) at (120:2.5) [v-tiny] {};
        \node (rrf) at (240:2.5) [v-tiny] {};
        \node (e) at (0:5.5) [v-tiny] {};
        \node (rr) at (240:5.5) [v-tiny] {};
        \node (r) at (120:5.5) [v-tiny] {};
        \draw [r-thin] (e) to [bend right] (r);
        \draw [r-thin] (r) to [bend right] (rr);
        \draw [r-thin] (rr) to [bend right] (e);
        \draw [r-thin] (f) to [bend left] (rrf);
        \draw [r-thin] (rrf) to [bend left] (rf);
        \draw [r-thin] (rf) to [bend left] (f);
        \draw [bb-thin] (e) to [bend left] (rrf);
        \draw [bb-thin] (rr) to [bend left] (rf);
        \draw [bb-thin] (r) to [bend left] (f);
        \node (o-in_r2) at (165:4.8) {};
        \node (o-out_r2) at (330:4.8) {};
        \node (p_r2) at (60:4.8) {};
      \end{scope}
      \draw [o] (o-out_1) to[bend right=45] (o-in_r);
      \draw [o] (o-out_r) to[bend right=45] (o-in_r2);
      \draw [o] (o-out_r2) to[bend right=45] (o-in_1);
      \draw [o] (o-out_f) to[bend left=30] (o-in_r2f);
      \draw [o] (o-out_r2f) to[bend left=35] (o-in_rf);
      \draw [o] (o-out_rf) to[bend left=30] (o-in_f);
      \draw [p] (p_1) to (p_f);
      \draw [p] (p_r) to (p_rf);
      \draw [p] (p_r2) to (p_r2f);
    \end{scope}
    %%%
    %%%%
    %%%%%=============================================================
    %%%%
   \begin{scope}[shift={(10,0)}]
    \begin{scope}[shift={(0:1.25)},scale=.12]
      \draw[grey, fill=lightgrey] (0,0) circle (60mm);
      \node (e) at (90:5.5) [v-tiny] {};
      \node (rrf) at (30:5.5) [v-tiny] {};
      \node (rr) at (-30:5.5) [v-tiny] {};
      \node (rf) at (-90:5.5) [v-tiny] {};
      \node (r) at (-150:5.5) [v-tiny] {};
      \node (f) at (150:5.5) [v-tiny] {};
      \path[gg-thin] (e) to (rf);
      \path[bb-thin] (rf) to (r);
      \path[gg-thin] (r) to (rrf);
      \path[bb-thin] (rrf) to (rr);
      \path[gg-thin] (rr) to (f);
      \path[bb-thin] (f) to (e);
      \node (o-in_f) at (90:4.8) {};
      \node (o-out_f) at (270:4.8) {};
      \node (p_f) at (0:4.8) {};
    \end{scope}
    %%
    \begin{scope}[shift={(120:1.25)},scale=.12]
      \draw[grey, fill=lightgrey] (0,0) circle (60mm);
      \node (e) at (90:5.5) [v-tiny] {};
      \node (rrf) at (30:5.5) [v-tiny] {};
      \node (rr) at (-30:5.5) [v-tiny] {};
      \node (rf) at (-90:5.5) [v-tiny] {};
      \node (r) at (-150:5.5) [v-tiny] {};
      \node (f) at (150:5.5) [v-tiny] {};
      \path[bb-thin] (e) to (rrf);
      \path[bb-thin] (r) to (f);
      \path[bb-thin] (rr) to (rf);
      \path[gg-thin] (e) to (f);
      \path[gg-thin] (r) to (rf);
      \path[gg-thin] (rr) to (rrf);
      \node (o-in_rf) at (195:4.8) {};
      \node (o-out_rf) at (30:4.8) {};
      \node (p_rf) at (120:4.8) {};
    \end{scope}
    %%
    \begin{scope}[shift={(240:1.25)},scale=.12]
      \draw[grey, fill=lightgrey] (0,0) circle (60mm);
      \node (e) at (90:5.5) [v-tiny] {};
      \node (rrf) at (30:5.5) [v-tiny] {};
      \node (rr) at (-30:5.5) [v-tiny] {};
      \node (rf) at (-90:5.5) [v-tiny] {};
      \node (r) at (-150:5.5) [v-tiny] {};
      \node (f) at (150:5.5) [v-tiny] {};
      \path[bb-thin] (e) to (rf);
      \path[bb-thin] (r) to (rrf);
      \path[bb-thin] (rr) to (f);
      \path[gg-thin] (e) to (rrf);
      \path[gg-thin] (r) to (f);
      \path[gg-thin] (rr) to (rf);
      \node (o-in_r2f) at (330:4.8) {};
      \node (o-out_r2f) at (165:4.8) {};
      \node (p_r2f) at (240:4.8) {};
    \end{scope}
    %%
    %%----------------------------------------------------------------
    %%
    \begin{scope}[shift={(0:3.5)},scale=.12]
      \draw[grey, fill=lightgrey] (0,0) circle (60mm);
      \node at (0,0) {$id$};
      \node (e) at (90:5.5) [v-tiny] {};
      \node (rrf) at (30:5.5) [v-tiny] {};
      \node (rr) at (-30:5.5) [v-tiny] {};
      \node (rf) at (-90:5.5) [v-tiny] {};
      \node (r) at (-150:5.5) [v-tiny] {};
      \node (f) at (150:5.5) [v-tiny] {};
      \path[gg-thin] (e) to (rrf);
      \path[bb-thin] (rrf) to (rr);
      \path[gg-thin] (rr) to (rf);
      \path[bb-thin] (rf) to (r);
      \path[gg-thin] (r) to (f);
      \path[bb-thin] (f) to (e);
      \node (o-in_1) at (270:4.8) {};
      \node (o-out_1) at (90:4.8) {};
       \node (p_1) at (180:4.8) {};
    \end{scope}
    %%
    \begin{scope}[shift={(120:3.5)},scale=.12]
      \draw[grey, fill=lightgrey] (0,0) circle (60mm);
      \node (e) at (90:5.5) [v-tiny] {};
      \node (rrf) at (30:5.5) [v-tiny] {};
      \node (rr) at (-30:5.5) [v-tiny] {};
      \node (rf) at (-90:5.5) [v-tiny] {};
      \node (r) at (-150:5.5) [v-tiny] {};
      \node (f) at (150:5.5) [v-tiny] {};
      \path[gg-thin] (e) to (f);
      \path[gg-thin] (r) to (rf);
      \path[gg-thin] (rr) to (rrf);
      \path[bb-thin] (e) to (rf);
      \path[bb-thin] (r) to (rrf);
      \path[bb-thin] (rr) to (f);
       \node (o-in_r) at (30:4.8) {};
      \node (o-out_r) at (195:4.8) {};
       \node (p_r) at (300:4.8) {};
    \end{scope}
    %%
    \begin{scope}[shift={(240:3.5)},scale=.12]
      \draw[grey, fill=lightgrey] (0,0) circle (60mm);
      \node (e) at (90:5.5) [v-tiny] {};
      \node (rrf) at (30:5.5) [v-tiny] {};
      \node (rr) at (-30:5.5) [v-tiny] {};
      \node (rf) at (-90:5.5) [v-tiny] {};
      \node (r) at (-150:5.5) [v-tiny] {};
      \node (f) at (150:5.5) [v-tiny] {};
      \path[gg-thin] (e) to (rf);
      \path[gg-thin] (r) to (rrf);
      \path[gg-thin] (rr) to (f);
      \path[bb-thin] (e) to (rrf);
      \path[bb-thin] (r) to (f);
      \path[bb-thin] (rr) to (rf);
      \node (o-in_r2) at (165:4.8) {};
      \node (o-out_r2) at (330:4.8) {};
      \node (p_r2) at (60:4.8) {};
    \end{scope}
    \draw [o] (o-out_1) to[bend right=45] (o-in_r);
    \draw [o] (o-out_r) to[bend right=45] (o-in_r2);
    \draw [o] (o-out_r2) to[bend right=45] (o-in_1);
    \draw [o] (o-out_f) to[bend left=30] (o-in_r2f);
    \draw [o] (o-out_r2f) to[bend left=35] (o-in_rf);
    \draw [o] (o-out_rf) to[bend left=30] (o-in_f);
    \draw [p] (p_1) to (p_f);
    \draw [p] (p_r) to (p_rf);
    \draw [p] (p_r2) to (p_r2f);
    \end{scope}
  \end{tikzpicture}
  \]
  In this problem, we will analyze the automorphism group of $D_4$,
  and construct similar Cayley diagrams.
  \begin{enumerate}
  \item The inner automorphism group of $D_4$ is
    \[
    \Inn(D_4)=\big\{\Id,\varphi_r,\varphi_f,\varphi_{r\!f}\big\}
    \cong D_4/Z(D_4)=D_4/\<r^2\>\cong V_4,
    \]    
    and additionally, there is a nontrivial outer automorphism
    \[
    \alpha\colon D_4\longto D_4,\qquad \alpha(r)=r,\quad \alpha(f)=rf. 
    \]
    that swaps the ``two types'' of reflections of the
    square. Construct a Cayley diagram for $\Aut(D_4)$, as the
    semidirect product of $\Inn(D_4)=\<\varphi_r,\varphi_f\>$ and
    $\Out(D_4)=\<\alpha\>$. Label each node with the product of an
    element of $\Inn(D_4)$ and an element of $\Out(D_4)$.  The
    previous problem should be very helpful.

    
  \item Determine which elements from the previous part are the following
    automorphisms, and whether or not they are inner.

    \vspace{-6mm}
    
    \tikzstyle{v-tiny} = [circle, draw, fill=lightgrey,inner sep=0pt,
      minimum size=3mm]
    \tikzstyle{r-out-r} = [draw, very thick, Red,-stealth,bend right=33]
    \tikzstyle{r-in-l} = [draw, very thick, Red,-stealth,bend left=25]
    \tikzstyle{r-out-l} = [draw, very thick, Red,-stealth,bend left=33]
    \tikzstyle{r-in-r} = [draw, very thick, Red,-stealth,bend right=25]
    \[
    \hspace*{-10mm}
    \begin{tikzpicture}[scale=.65]
      \begin{scope}[shift={(0,0)}]
        \node (1) at (0:2) [v-tiny] {\scriptsize $1$};
        \node (r) at (90:2) [v-tiny] {};
        \node (r2) at (180:2) [v-tiny] {};
        \node (r3) at (270:2) [v-tiny] {};
        \node (f) at (0:1) [v-tiny] {};
        \node (r3f) at (270:1) [v-tiny] {};
        \node (r2f) at (180:1) [v-tiny] {};
        \node (rf) at (90:1) [v-tiny] {};
        \draw [r-out-r] (1) to (r);
        \draw [r-out-r] (r) to (r2);
        \draw [r-out-r] (r2) to (r3);
        \draw [r-out-r] (r3) to (1);
        \draw [r-in-l] (f) to (r3f);
        \draw [r-in-l] (r3f) to (r2f);
        \draw [r-in-l] (r2f) to (rf);
        \draw [r-in-l] (rf) to (f);
        \draw [bb] (1) to (f);
        \draw [bb] (r) to (rf);
        \draw [bb] (r2) to (r2f);
        \draw [bb] (r3) to (r3f);
        \draw[-stealth'] (2.6,0) -- (3.9,0); %node[midway,above] {$\gamma$};
      \end{scope}
      %%
      \begin{scope}[shift={(6.5,0)},scale=1]
        \tikzstyle{r-out} = [draw, very thick, Red,-stealth,bend right=33]
        \tikzstyle{r-in} = [draw, very thick, Red,-stealth,bend left=25]
        \node (1) at (0:2) [v-tiny] {\scriptsize $1$};
        \node (r) at (90:2) [v-tiny] {};
        \node (r2) at (180:2) [v-tiny] {};
        \node (r3) at (270:2) [v-tiny] {};
        \node (f) at (0:1) [v-tiny] {};
        \node (r3f) at (270:1) [v-tiny] {};
        \node (r2f) at (180:1) [v-tiny] {};
        \node (rf) at (90:1) [v-tiny] {};
        \draw [r-out-l] (r) to (1);
        \draw [r-out-l] (r2) to (r);
        \draw [r-out-l] (r3) to (r2);
        \draw [r-out-l] (1) to (r3);
        \draw [r-in-r] (r3f) to (f);
        \draw [r-in-r] (r2f) to (r3f);
        \draw [r-in-r] (rf) to (r2f);
        \draw [r-in-r] (f) to (rf);
        \draw [bb] (1) to (f);
        \draw [bb] (r) to (rf);
        \draw [bb] (r2) to (r2f);
        \draw [bb] (r3) to (r3f);
      \end{scope}
      \begin{scope}[shift={(13.5,0)},scale=1]
        \node (1) at (0:2) [v-tiny] {\scriptsize $1$};
        \node (s) at (45:2) [v-tiny] {};
        \node (st) at (90:2) [v-tiny] {};
        \node (sts) at (135:2) [v-tiny] {};
        \node (stst) at (180:2) [v-tiny] {};
        \node (tst) at (225:2) [v-tiny] {};
        \node (ts) at (270:2) [v-tiny] {};
        \node (t) at (315:2) [v-tiny] {};
        \draw [bb] (1) to (s); \draw [gg] (s) to (st);
        \draw [bb] (st) to (sts); \draw [gg] (sts) to (stst);
        \draw [bb] (stst) to (tst); \draw [gg] (tst) to (ts);
        \draw [bb] (ts) to (t); \draw [gg] (t) to (1); 
        \draw[-stealth'] (2.6,0) -- (3.9,0);
      \end{scope}
      \begin{scope}[shift={(20,0)},scale=1]
        \node (1) at (0:2) [v-tiny] {\scriptsize $1$};
        \node (s) at (45:2) [v-tiny] {};
        \node (st) at (90:2) [v-tiny] {};
        \node (sts) at (135:2) [v-tiny] {};
        \node (stst) at (180:2) [v-tiny] {};
        \node (tst) at (225:2) [v-tiny] {};
        \node (ts) at (270:2) [v-tiny] {};
        \node (t) at (315:2) [v-tiny] {};
        \draw [gg] (1) to (s); \draw [bb] (s) to (st);
        \draw [gg] (st) to (sts); \draw [bb] (sts) to (stst);
        \draw [gg] (stst) to (tst); \draw [bb] (tst) to (ts);
        \draw [gg] (ts) to (t); \draw [bb] (t) to (1); 
      \end{scope}
    \end{tikzpicture}
    \]
  \item As was done above for $\Aut(D_3)$, create the Cayley diagram
    for $\Aut(D_4)$, and label the nodes with re-wired copies of the
    Cayley diagram of $D_4=\<\Alert{r},\Balert{f}\>$. Do this for both
    of the following Cayley diagrams of $\Aut(D_4)$ shown below.
        
    \tikzstyle{v-tiny} = [circle, draw, fill=lightgrey,inner sep=0pt,
      minimum size=4.5mm]
    \tikzstyle{o-out} = [draw, very thick, orange,-stealth,bend right=33]
    \tikzstyle{o-in} = [draw, very thick, orange,-stealth,bend left=25]
    \tikzstyle{rr-faded} = [draw, very thick, lightred]
    \tikzstyle{bb-faded} = [draw, very thick, lightblue]
    \tikzstyle{gg-faded} = [draw, very thick, lightgreen]
    \tikzstyle{yy-faded} = [draw, very thick, Gold!30!white]
    \tikzstyle{pp-faded} = [draw, very thick, amethyst!30!white]
    \tikzstyle{tt-faded} = [draw, very thick, turq!30!white]
    \[
    \hspace*{-15mm}
    \begin{tikzpicture}[scale=1.25]
      \begin{scope}[shift={(0,0)}]
        \node (1) at (0:2) [v-tiny] {};
        \node (r) at (90:2) [v-tiny] {};
        \node (r2) at (180:2) [v-tiny] {};
        \node (r3) at (270:2) [v-tiny] {};
        \node (f) at (0:1) [v-tiny] {};
        \node (r3f) at (270:1) [v-tiny] {};
        \node (r2f) at (180:1) [v-tiny] {};
        \node (rf) at (90:1) [v-tiny] {};
        \draw [o-out] (1) to (r);
        \draw [o-out] (r) to (r2);
        \draw [o-out] (r2) to (r3);
        \draw [o-out] (r3) to (1);
        \draw [o-in] (f) to (r3f);
        \draw [o-in] (r3f) to (r2f);
        \draw [o-in] (r2f) to (rf);
        \draw [o-in] (rf) to (f);
        \draw [pp] (1) to (f);
        \draw [pp] (r) to (rf);
        \draw [pp] (r2) to (r2f);
        \draw [pp] (r3) to (r3f);
      \end{scope}
      %%
      \begin{scope}[shift={(5,-2)},scale=2.5]
        \node [v-tiny] (100) at (0,0) {};
        \node [v-tiny] (101) at (0,1) {};
        \node [v-tiny] (111) at (1,1) {};
        \node [v-tiny] (110) at (1,0) {};
        \node [v-tiny] (000) at (.5,.5) {};
        \node [v-tiny] (001) at (.5,1.5) {};
        \node [v-tiny] (011) at (1.5,1.5) {};
        \node [v-tiny] (010) at (1.5,.5) {};
        \draw [tt-faded] (000) to (001);
        \draw [pp-faded] (000) to (010);
        \draw [yy-faded] (100) to (000);
        \draw [pp] (100) to (101); \draw [pp] (110) to (111);
        \draw [tt] (010) to (011);
        \draw [tt] (100) to (110); \draw [tt] (101) to (111);
        \draw [pp] (001) to (011);
        \draw [yy] (111) to (011); \draw [yy] (101) to (001); 
        \draw [yy] (110) to (010);
      \end{scope}
    \end{tikzpicture}
    \]

  \item Repeat the previous part using the Cayley diagram of
    $D_4=\<\Balert{s},\Galert{t}\>=\<\Balert{f},\Galert{rf}\>$.
    
  \end{enumerate}
  
\end{enumerate}

\end{document}

