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\tikzstyle{R} = [draw, very thick, Red, -stealth']        % Red --->
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\pagestyle{fancy} \lhead{{\sf Homework 11 $|$ Due April 11 (Monday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Spring 2022 $|$ M.~Macauley}}

\begin{document}

%$\;$

\begin{enumerate}

  %%--------------------------------------------------------------------
  
\item In this problem, we will explore the actions of the dicyclic
  group $\Dic_6$ and its automorphism group on itself and its
  subgroups by conjugation. A Cayley diagram, subgroup lattice, and
  conjugacy classes are shown below.
  \[
  \tikzstyle{R6-out} = [draw, very thick, Red,-stealth,bend right=18]
  \tikzstyle{R6-in} = [draw, very thick, Red,-stealth,bend left=12]
  \tikzstyle{R-out} = [draw, very thick, Red,-stealth,bend right=15]
  \tikzstyle{R-in} = [draw, very thick, Red,-stealth,bend left=12]
  \tikzstyle{B} = [draw, very thick, Blue,-stealth,bend right=25]
   \tikzstyle{v-small} = [circle, draw, fill=lightgrey,inner sep=0pt,
    minimum size=6mm]
   \hspace*{-6mm}
   \begin{tikzpicture}[scale=1.25]
    \begin{scope}[shift={(0,2.1)},scale=1.05]
      \tikzstyle{every node}=[font=\small]
      \node (1) at (0:2) [v-small] {$1$};
      \node (r) at (60:2) [v-small] {$r$};
      \node (r2) at (120:2) [v-small] {$r^2$};
      \node (r3) at (180:2) [v-small] {$r^3$};
      \node (r4) at (240:2) [v-small] {$r^4$};
      \node (r5) at (300:2) [v-small] {$r^5$};
      \node (s) at (0:1) [v-small] {$s$};
      \node (sr5) at (60:1) [v-small] {$rs$};
      \node (sr4) at (120:1) [v-small] {$r^2\!s$};
      \node (sr3) at (180:1) [v-small] {$r^3\!s$};
      \node (sr2) at (240:1) [v-small] {$r^4\!s$};
      \node (sr) at (300:1) [v-small] {$r^5\!s$};
      %%
      \path[b] (1) to (s);
      \path[B] (s) to (r3);
      \path[b] (r3) to (sr3);
      \path[B] (sr3) to (1);
      %%
      \path[b] (r) to (sr5);
      \path[B] (sr5) to (r4);
      \path[b] (r4) to (sr2);
      \path[B] (sr2) to (r);
      %%
      \path[b] (r2) to (sr4);
      \path[B] (sr4) to (r5);
      \path[b] (r5) to (sr);
      \path[B] (sr) to (r2);
      \path[R6-out] (1) to (r);
      \path[R6-out] (r) to (r2);
      \path[R6-out] (r2) to (r3);
      \path[R6-out] (r3) to (r4);
      \path[R6-out] (r4) to (r5);
      \path[R6-out] (r5) to (1);
      \path[R-in] (sr5) to (s);
      \path[R-in] (sr4) to (sr5);
      \path[R-in] (sr3) to (sr4);
      \path[R-in] (sr2) to (sr3);
      \path[R-in] (sr) to (sr2);
      \path[R-in] (s) to (sr);
      %%
      \node at (0,0) {\small $\Dic_6$};
    \end{scope}
    %%%%
    \begin{scope}[shift={(4.5,0)},scale=.7,shorten >= -2pt, shorten <= -2pt]
      \tikzstyle{every node}=[font=\normalsize]
      \node(G) at (0,6) {$\Dic_6$};
      \node(b) at (-1.5,4.5) {$\<r\>$};
      \node(aa) at (0,1.5) {$\<r^3\>$};
      \node(bb) at (-2,2.25) {$\<r^2\>$};
      \node(a) at (.5,24/7) {$\<s\>$};
      \node(ab) at (1.75,24/7) { $\<rs\>$};
      \node(ba) at (3,24/7) {$\<r^2\!s\>$};
      \node(1) at (0,0) {$\left\<1\right\>$};
      %\draw(1)--(aa); \draw(1)--(bb);
      %\draw(b)--(aa); \draw(b)--(bb);
      %\draw(G)--(a); \draw(G)--(ab); \draw(G)--(ba); \draw(G)--(b);
      %\draw(aa)--(a); \draw(aa)--(ab); \draw(aa)--(ba);
      \draw (1) to (aa);
      \draw (1) to (bb);
      \draw (b) to (aa);
      \draw (b) to (bb);
      \draw (aa) to (a);
      \draw (aa) to (ab);
      \draw (aa) to (ba);
      \draw (G) to (a);
      \draw (G) to (ab);
      \draw (G) to (ba);
      \draw (G) to (b);   
      %\node at (0,-1) {$\Dic_{12}=\<a,b\mid a^4=b^3,\;bab=a\>$};
    \end{scope}
    \begin{scope}[shift={(7.35,.6)}]
      \tikzstyle{every node}=[font=\small]
      \draw[thin,fill=conj-purple] (0,0) rectangle (1,1.6);
      \draw[thin,fill=conj-orange] (1,0) rectangle (3,1.6);
      \draw[thin,fill=conj-orange] (0,1.6) rectangle (3,3.2);
      \draw[thin] (1,0) to (1,1.6);
      \draw[thin] (0,2.4) to (3,2.4);
      \draw[thin] (0,.8) to (3,.8);
      \node at (.5,.4) {$1$};
      \node at (1.5,.4) {$r$};
      \node at (2.5,.4) {$r^5$};
      \node at (.5,1.2) {$r^3$};
      \node at (1.5,1.2) {$r^2$};
      \node at (2.5,1.2) {$r^4$};
      \node at (.5,2) {$s$};
      \node at (1.5,2) {$r^2s$};
      \node at (2.5,2) {$r^4s$};
      \node at (.5,2.8) {$rs$};
      \node at (1.5,2.8) {$r^5s$};
      \node at (2.5,2.8) {$r^3s$};
      %\node at (1.5,-.6) {\small conjugacy classes};
    \end{scope}
    %%%
  \end{tikzpicture}
  \]

  \begin{enumerate}
  \item The right action of $\Dic_6$ on itself by conjugation is
    defined by the homomorphism
    \[
    \phi\colon\Dic_6\longto\Perm(S)\,,\qquad \phi(g)=\mbox{\footnotesize the
      permutation that sends each $x\mapsto g^{-1}xg$.}
    \]
    Draw the action diagram and construct the fixed point table. Find
    $\stab(s)$ for each $s\in S$, $\fix(g)$ for each $g\in G$, as well as
    $\Ker(\phi)$ and $\Fix(\phi)$.
  \item The inner automorphism group
    \[
    \Inn(\Dic_6)\cong\Dic_6/Z(\Dic_6)=\Dic_6/\<r^3\>\cong D_3
    \]
    acts on $\Dic_6$, and the action diagram is the same as the
    one in Part~(a). Construct the fixed point table of this
    action and find $\stab(s)$, $\fix(g)$, $\Ker(\phi)$ and
    $\Fix(\phi)$. Then draw the subgroup lattice of
    $\Inn(\Dic_6)=\<\varphi_r,\varphi_s\>$, where $\varphi_g\colon
    x\mapsto g^{-1}xg$.
      \item The automorphism group of $\Dic_6$ is
    $\Aut(\Dic_6)=\<\varphi_r,\varphi_s,\alpha\>$, where $\alpha$ is
    the outer automorphism defined by
    \[
    \alpha\colon\Dic_6\longto\Dic_6,\qquad\alpha(r)=r,\quad\alpha(s)=s^{-1}=r^3s,
    \]
    that ``reverses'' the blue arrows. Construct the action diagram,
    fixed point table, and find $\stab(s)$, $\fix(g)$, $\Ker(\phi)$
    and $\Fix(\phi)$.
  \item What familiar group is
    $\Aut(\Dic_6)=\<\varphi_r,\varphi_s,\alpha\>$ isomorphic to?
    Construct a Cayley diagram and subgroup lattice.
  \item The group $\Aut(\Dic_6)$ also acts on the conjugacy classes of
    $\Dic_6$. Construct the action diagram, fixed point table, and
    find $\stab(s)$, $\fix(g)$, $\Ker(\phi)$ and $\Fix(\phi)$.
  \item The group $\Dic_6$ acts on its subgroups by conjugation, via
    the homomorphism
    \[
    \phi\colon\Dic_6\longto\Perm(S)\,,\qquad \phi(g)=\mbox{\footnotesize the
      permutation that sends each $H\mapsto g^{-1}Hg$.}
    \]
    Construct the action diagram superimposed on the subgroup
    lattice. Then construct the fixed point table and find $\stab(s)$,
    $\fix(g)$, $\Ker(\phi)$ and $\Fix(\phi)$.
    
  \end{enumerate}

\item Carry out the following steps for the groups $C_7\rtimes C_3$
  and $C_9\rtimes C_3$, whose Cayley diagrams are shown below.
  \tikzstyle{R7} = [draw, very thick, Red,-stealth,bend right=15]
  \tikzstyle{R9} = [draw, very thick, Red,-stealth,bend right=10]
  \tikzstyle{R2} = [draw, very thick, Red,-stealth,bend right=20]
  \tikzstyle{R3} = [draw, very thick, Red,-stealth,bend left=8]
  \tikzstyle{v-tiny} = [circle, draw, fill=lightgrey,inner sep=0pt,
    minimum size=6mm] \tikzstyle{v-small} = [circle, draw,
    fill=lightgrey,inner sep=0pt, minimum size=5.5mm]
  \tikzstyle{v-big} = [circle, draw, fill=lightgrey,inner sep=0pt,
    minimum size=6mm]
  \[
  \hspace*{-12mm}
  \begin{tikzpicture}[scale=1.25]
    \tikzstyle{every node}=[font=\scriptsize]
    \begin{scope}[shift={(0,0)},scale=1]
      \node (s2) at (0:1) [v-small] {$s^2$};
      \node (rs2) at (51.45:1) [v-small] {$rs^2$};
      \node (r2s2) at (102.9:1) [v-small] {$r^2\!s^2$};
      \node (r3s2) at (154.3:1) [v-small] {$r^3\!s^2$};
      \node (r4s2) at (205.7:1) [v-small] {$r^4\!s^2$};
      \node (r5s2) at (257.1:1) [v-small] {$r^5\!s^2$};
      \node (r6s2) at (308.6:1) [v-small] {$r^6\!s^2$};
      %%
      \node (s) at (0:2) [v-small] {$s$};
      \node (rs) at (51.45:2) [v-small] {$rs$};
      \node (r2s) at (102.9:2) [v-small] {$r^2\!s$};
      \node (r3s) at (154.3:2) [v-small] {$r^3\!s$};
      \node (r4s) at (205.7:2) [v-small] {$r^4\!s$};
      \node (r5s) at (257.1:2) [v-small] {$r^5\!s$};
      \node (r6s) at (308.6:2) [v-small] {$r^6\!s$};
      %%
      \node (1) at (0:3) [v-small] {$1$};
      \node (r) at (51.45:3) [v-small] {$r$};
      \node (r2) at (102.9:3) [v-small] {$r^2$};
      \node (r3) at (154.3:3) [v-small] {$r^3$};
      \node (r4) at (205.7:3) [v-small] {$r^4$};
      \node (r5) at (257.1:3) [v-small] {$r^5$};
      \node (r6) at (308.6:3) [v-small] {$r^6$};
      %%
      \draw [R7] (1) to (r);
      \draw [R7] (r) to (r2);
      \draw [R7] (r2) to (r3);
      \draw [R7] (r3) to (r4);
      \draw [R7] (r4) to (r5);
      \draw [R7] (r5) to (r6);
      \draw [R7] (r6) to (1);
      %%
      \draw [R2] (s) to (r2s);
      \draw [R2] (r2s) to (r4s);
      \draw [R2] (r4s) to (r6s);
      \draw [R2] (r6s) to (rs);
      \draw [R2] (rs) to (r3s);
      \draw [R2] (r3s) to (r5s);
      \draw [R2] (r5s) to (s);
      %%
      \draw [r] (s2) to (r4s2);
      \draw [r] (r4s2) to (rs2);
      \draw [r] (rs2) to (r5s2);
      \draw [r] (r5s2) to (r2s2);
      \draw [r] (r2s2) to (r6s2);
      \draw [r] (r6s2) to (r3s2);
      \draw [r] (r3s2) to (s2);
      %%
      \draw [b] (1) to (s); \draw [b] (s) to (s2);
      \draw [b,bend left] (s2) to (1);
      \draw [b] (r) to (rs); \draw [b] (rs) to (rs2);
      \draw [b,bend left] (rs2) to (r);
      \draw [b] (r2) to (r2s); \draw [b] (r2s) to (r2s2);
      \draw [b,bend left] (r2s2) to (r2);
      \draw [b] (r3) to (r3s); \draw [b] (r3s) to (r3s2);
      \draw [b,bend left] (r3s2) to (r3);
      \draw [b] (r4) to (r4s); \draw [b] (r4s) to (r4s2);
      \draw [b,bend left] (r4s2) to (r4);
      \draw [b] (r5) to (r5s); \draw [b] (r5s) to (r5s2);
      \draw [b,bend left] (r5s2) to (r5);
      \draw [b] (r6) to (r6s); \draw [b] (r6s) to (r6s2);
      \draw [b,bend left] (r6s2) to (r6);
    \end{scope}
    %%
    \begin{scope}[shift={(7,0)}]
      \node (1) at (0:3) [v-small] {$1$};
      \node (r) at (40:3) [v-small] {$r$};
      \node (r2) at (80:3) [v-small] {$r^2$};
      \node (r3) at (120:3) [v-small] {$r^3$};
      \node (r4) at (160:3) [v-small] {$r^4$};
      \node (r5) at (200:3) [v-small] {$r^5$};
      \node (r6) at (240:3) [v-small] {$r^6$};
      \node (r7) at (280:3) [v-small] {$r^7$};
      \node (r8) at (320:3) [v-small] {$r^8$};
      %%
      \node (s) at (0:2) [v-small] {$s$};
      \node (rs) at (40:2) [v-small] {$rs$};
      \node (r2s) at (80:2) [v-small] {$r^2\!s$};
      \node (r3s) at (120:2) [v-small] {$r^3\!s$};
      \node (r4s) at (160:2) [v-small] {$r^4\!s$};
      \node (r5s) at (200:2) [v-small] {$r^5\!s$};
      \node (r6s) at (240:2) [v-small] {$r^6\!s$};
      \node (r7s) at (280:2) [v-small] {$r^7\!s$};
      \node (r8s) at (320:2) [v-small] {$r^8\!s$};
      %%
      \node (s2) at (0:1) [v-small] {$s^2$};
      \node (rs2) at (40:1) [v-small] {$rs^2$};
      \node (r2s2) at (80:1) [v-small] {$r^2\!s^2$};
      \node (r3s2) at (120:1) [v-small] {$r^3\!s^2$};
      \node (r4s2) at (160:1) [v-small] {$r^4\!s^2$};
      \node (r5s2) at (200:1) [v-small] {$r^5\!s^2$};
      \node (r6s2) at (240:1) [v-small] {$r^6\!s^2$};
      \node (r7s2) at (280:1) [v-small] {$r^7\!s^2$};
      \node (r8s2) at (320:1) [v-small] {$r^8\!s^2$};
      %%
      \draw [R9] (1) to (r);
      \draw [R9] (r) to (r2);
      \draw [R9] (r2) to (r3);
      \draw [R9] (r3) to (r4);
      \draw [R9] (r4) to (r5);
      \draw [R9] (r5) to (r6);
      \draw [R9] (r6) to (r7);
      \draw [R9] (r7) to (r8);
      \draw [R9] (r8) to (1);
      %%
      \draw [R3] (s) to (r7s);
      \draw [R3] (r7s) to (r5s);
      \draw [R3] (r5s) to (r3s);
      \draw [R3] (r3s) to (rs);
      \draw [R3] (rs) to (r8s);
      \draw [R3] (r8s) to (r6s);
      \draw [R3] (r6s) to (r4s);
      \draw [R3] (r4s) to (r2s);
      \draw [R3] (r2s) to (s);
      %%
      \draw [r] (s2) to (r4s2);
      \draw [r] (r4s2) to (r8s2);
      \draw [r] (r8s2) to (r3s2);
      \draw [r] (r3s2) to (r7s2);
      \draw [r] (r7s2) to (r2s2);
      \draw [r] (r2s2) to (r6s2);
      \draw [r] (r6s2) to (rs2);
      \draw [r] (rs2) to (r5s2);
      \draw [r] (r5s2) to (s2);
      %%
      \draw [b] (1) to (s); \draw [b] (s) to (s2);
      \draw [b,bend left] (s2) to (1);
      \draw [b] (r) to (rs); \draw [b] (rs) to (rs2);
      \draw [b,bend left] (rs2) to (r);
      \draw [b] (r2) to (r2s); \draw [b] (r2s) to (r2s2);
      \draw [b,bend left] (r2s2) to (r2);
      \draw [b] (r3) to (r3s); \draw [b] (r3s) to (r3s2);
      \draw [b,bend left] (r3s2) to (r3);
      \draw [b] (r4) to (r4s); \draw [b] (r4s) to (r4s2);
      \draw [b,bend left] (r4s2) to (r4);
      \draw [b] (r5) to (r5s); \draw [b] (r5s) to (r5s2);
      \draw [b,bend left] (r5s2) to (r5);
      \draw [b] (r6) to (r6s); \draw [b] (r6s) to (r6s2);
      \draw [b,bend left] (r6s2) to (r6);
      \draw [b] (r7) to (r7s); \draw [b] (r7s) to (r7s2);
      \draw [b,bend left] (r7s2) to (r7);
      \draw [b] (r8) to (r8s); \draw [b] (r8s) to (r8s2);
      \draw [b,bend left] (r8s2) to (r8);
    \end{scope}
  \end{tikzpicture}
  \]
  \begin{enumerate}
  \item Let $G$ act on its subgroups by conjugation. Draw the action
    diagram superimposed on the subgroup lattice. Construct the fixed
    point table, and find $\stab(H)$ for each $H\leq G$, $\Ker(\phi)$
    and $\Fix(\phi)$. Which collections of subgroups arise as
    $\fix(g)$ for some $g\in G$, and why?
  \item Let $G$ act on the right cosets of $H=\<s\>$, via the homomorphism
    \[
    \phi\colon G\longto\Perm(S)\,,\qquad \phi(g)=\mbox{\footnotesize the
      permutation that sends each $Hx\mapsto Hxg$.}
    \]
    Construct the action diagram, fixed point table, and find
    $\stab(Hx)$ for each right coset, $\Ker(\phi)$ and $\Fix(\phi)$. Which
    subsets of $S$ arise as $\fix(g)$ for some $g\in G$, and why? \bigskip
  \end{enumerate}


%%--------------------------------------------------------------------
  
\item Let $G$ be a group, not necessarily finite, and let $A$ and $B$
  be subgroups of finite index, but not necessarily normal (in
  particular, we cannot assume that $AB$ is a group).
  \begin{enumerate}
  \item Show that even if $AB$ is not a group, it is a disjoint union
    of cosets of $A$.
  \item Let $B$ act on
    $S=AB/A=\big\{Ax\mid x\in AB\big\}$ via the homomorphism
    \[
    \phi\colon G\longto\Perm(S)\,,\qquad \phi(g)=\mbox{\footnotesize the
      permutation that sends each $Ax\mapsto Axg$.}
  \]
  Show that there is only one orbit.
  \item Use the orbit-stabilizer theorem to show that $[A:A\cap B]=[AB:B]$. 
  \item Show that $[G:A\cap B]\leq[G:A][G:B]$. Give an explicit
    example of where the inequality is strict.
  \item Show that there is some $N\normaleq G$ contained in both $A$
    and $B$ with $[G:N]\leq\infty$.
  \end{enumerate}

  
\end{enumerate}

\end{document}



