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\usepackage{visualalgebra}  %% Put this *after* the TikZ packages

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\usepackage[numbers,square,sort&compress]{natbib}

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%\usepackage{enumerate}         % [shortlabels] option??
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\pagestyle{fancy} \lhead{{\sf Homework 9 $|$ Due March 14 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Visual Algebra $|$
    Spring 2025 $|$ M.~Macauley}}

\begin{document}

%$\;$

\begin{enumerate}

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

\item Let $A,B\leq G$ with $A$ normalizing $B$ (that is, $A\leq
  N_G(B)$, which implies that $AB\leq G$).
  \begin{enumerate}
  \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.] \medskip
  \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. [\emph{Hint}: Show that every
    commutator is trivial.]
  \item For the groups $\AGL_1(\Z_5)$, $\Dic_{10}$, $\SL_2(\Z_3)$, and
    $Q_8\rtimes C_9$, 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}
    \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
      $G^{(0)}:=G$, $G^{(1)}:=G'$, $G^{(2)}:=G''$, 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
      $G^{(i)}/G^{(i+1)}$ of each successive pair. \medskip
    \end{enumerate}
  \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 \emph{inner} (of the form $\varphi_x\colon
  g\mapsto x^{-1}gx$). Two Cayley graphs for
  $\Aut(D_3)$ are shown below.
%  because $\Inn(D_3)\normaleq\Aut(D_3)$ and
%  $\Inn(D_3)\cong D/Z(D_3)\cong D_3$.


  \vspace{-4mm}
  
  \tikzstyle{v-tiny} = [circle, draw, fill=gray,inner sep=0pt, minimum size=1.2mm] 
  \tikzstyle{gg-thin} = [draw, eGreen]
  \tikzstyle{r-thin} = [draw, eRed, -stealth]
  \tikzstyle{bb-thin} = [draw, eBlue]
  \tikzstyle{o} = [draw, very thick,eOrange, -stealth]
  \tikzstyle{p} = [draw, very thick,ePurple]
  %%
  \[
  \hspace*{-10mm}
  \begin{tikzpicture}[scale=.95]
    %%%%
    %%%%%=============================================================
    %%%%
    \begin{scope}[shift={(0,0)},scale=1]
      \begin{scope}[shift={(0:1.25)},scale=.12]
        \draw[gray, fill=lightgray] (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[gray, fill=lightgray] (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[gray, fill=lightgray] (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[gray, fill=lightgray] (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[gray, fill=lightgray] (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[gray, fill=lightgray] (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[gray, fill=lightgray] (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[gray, fill=lightgray] (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[gray, fill=lightgray] (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[gray, fill=lightgray] (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[gray, fill=lightgray] (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[gray, fill=lightgray] (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 $D_4$,
  construct analogous Cayley graphs for $\Aut(D_4)\cong D_4$.
  \begin{enumerate}
  \item For each of the Cayley graphs of $\Aut(D_4)$ shown below,
    label the nodes with rewired copies of the Cayley graph of
    $D_4=\<\Alert{r},\Balert{f}\>$.
        
        



    \[
    \hspace*{-15mm}
    \begin{tikzpicture}[scale=1.25]
    \definecolor{Gold}{rgb}{.83,.67,.08}
    \definecolor{turq}{rgb}{0.12 0.69 0.71}
    \tikzstyle{tt} = [draw, very thick, turq]      % Turquoise ----
    \tikzstyle{yy} = [draw, very thick, Gold]
    \tikzstyle{yFaded} = [draw, very thick, Gold!30!white]
    \tikzstyle{tFaded} = [draw, very thick, turq!30!white]
    \tikzstyle{v} = [circle, draw, fill=lightgray,inner sep=0pt,
      minimum size=4.5mm]
    \tikzstyle{o-out} = [draw, very thick, eOrange,-stealth,bend right=33]
    \tikzstyle{o-in} = [draw, very thick, eOrange,-stealth,bend left=25]
    %%
    \begin{scope}[shift={(0,0)}]
      \node (1) at (0:2) [v] {\scriptsize Id};
      \node (r) at (90:2) [v] {};
      \node (r2) at (180:2) [v] {};
      \node (r3) at (270:2) [v] {};
      \node (f) at (0:1) [v] {};
      \node (r3f) at (270:1) [v] {};
      \node (r2f) at (180:1) [v] {};
      \node (rf) at (90:1) [v] {};
      \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);
      \node at (0,0) {$C_4\!\rtimes\!C_2$};
    \end{scope}
    %%
    \begin{scope}[shift={(5,-2)},scale=2.5]
      \node [v] (100) at (0,0) {};
      \node [v] (101) at (0,1) {};
      \node [v] (111) at (1,1) {\scriptsize Id};
      \node [v] (110) at (1,0) {};
      \node [v] (000) at (.5,.5) {};
      \node [v] (001) at (.5,1.5) {};
        \node [v] (011) at (1.5,1.5) {};
        \node [v] (010) at (1.5,.5) {};
        \draw [tFaded] (000) to (001);
        \draw [pFaded] (000) to (010);
        \draw [yFaded] (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);
        \node at (.75,.75) {$V_4\!\rtimes\!C_2$};
      \end{scope}
    \end{tikzpicture}
    \]

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

    
  \item Four of the eight automorphisms of $D_4$ are \emph{inner},
    which means they have the form $\varphi_x\colon g\mapsto x^{-1}gx$
    for some $x\in D_4$. In fact, the automorphism group of $D_4$ is
    isomorphic to the semidirect product
    \[
    \Aut(D_4)\cong\Inn(D_4)\rtimes C_2
    \]
    of the \emph{inner automorphism group}
    \[
    \Inn(D_4)=\big\{\Id,\varphi_r,\varphi_f,\varphi_{r\!f}\big\}
    \cong D_4/Z(D_4)=D_4/\<r^2\>\cong V_4
    \]    
    and the cyclic subgroup generated by an \emph{outer automorphism} of
    order $2$. In each of your Cayley graphs from Parts (a) and (b),
    label the nodes by the corresponding automorphism written as
    \[
    \Aut(D_4)=\big\{\Id,\,\varphi_r,\,\varphi_f,\,\varphi_{r\!f},\,
    \omega,\,\varphi_r\omega,\,\varphi_f\omega,\,\varphi_{r\!f}\omega\big\},
    \]
    where $\omega$ is the outer automorphism
    \[
    \omega\colon D_4\longto D_4,\qquad \alpha(r)=r,\quad \alpha(f)=rf
    \]
    of order $4$ that cyclically rotates axes of reflections of the square.
  \end{enumerate}

  \medskip
  
  %%--------------------------------------------------------------------

\item Construct each of the semidirect products below via our
  ``inflation process''.
  \begin{multicols}{4}
    \begin{enumerate}
    \item $D_3\times C_2$
    \item $D_3\rtimes C_2$
    \item $V_4\rtimes C_3$
    \item $C_3\rtimes V_4$.
    \end{enumerate}
  \end{multicols}
  Make sure you define the labeling maps $\theta\colon
  B\to\Aut(A)$. Then determine, with justification, what each group is
  isomorphic to. Use $D_3=\<a,b\mid a^2=b^2=(ab)^3=1\>$,
  $V_4=\<a,b\mid a^2=b^2=1\>$, and $C_n=\<c\mid c^n=1\>$ for the
  individual factors, and Cayley graphs corresponding to these
  generating sets.  

  %%---------------------------------------------------------------
  
\end{enumerate}

\end{document}    
    
  
 
